| L(s) = 1 | + 3.12e4·5-s − 2.84e5·7-s − 2.56e6·11-s + 8.73e6·13-s − 2.75e8·17-s − 2.49e8·19-s + 1.29e9·23-s + 7.32e8·25-s + 4.73e9·29-s − 7.74e9·31-s − 8.89e9·35-s − 2.31e10·37-s − 1.56e10·41-s − 8.13e9·43-s + 1.24e11·47-s + 4.06e10·49-s − 1.14e10·53-s − 8.02e10·55-s + 5.69e11·59-s − 7.77e11·61-s + 2.72e11·65-s + 6.56e10·67-s + 1.27e12·71-s − 2.26e12·73-s + 7.30e11·77-s − 1.30e11·79-s − 1.01e12·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.914·7-s − 0.437·11-s + 0.501·13-s − 2.76·17-s − 1.21·19-s + 1.82·23-s + 3/5·25-s + 1.47·29-s − 1.56·31-s − 0.817·35-s − 1.48·37-s − 0.513·41-s − 0.196·43-s + 1.69·47-s + 0.419·49-s − 0.0710·53-s − 0.390·55-s + 1.75·59-s − 1.93·61-s + 0.448·65-s + 0.0887·67-s + 1.18·71-s − 1.74·73-s + 0.399·77-s − 0.0603·79-s − 0.340·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(2.001397213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.001397213\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{6} T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 284528 T + 5764451730 p T^{2} + 284528 p^{13} T^{3} + p^{26} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 233472 p T + 211348177318 p^{2} T^{2} + 233472 p^{14} T^{3} + p^{26} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8733580 T + 171358494616206 T^{2} - 8733580 p^{13} T^{3} + p^{26} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 275179572 T + 38732394432870070 T^{2} + 275179572 p^{13} T^{3} + p^{26} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 249195440 T + 87493125117728118 T^{2} + 249195440 p^{13} T^{3} + p^{26} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1296464088 T + 1422530628910405102 T^{2} - 1296464088 p^{13} T^{3} + p^{26} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4734911532 T + 17143061080059851134 T^{2} - 4734911532 p^{13} T^{3} + p^{26} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7749542792 T + 44614833314679185598 T^{2} + 7749542792 p^{13} T^{3} + p^{26} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 23145039284 T + \)\(36\!\cdots\!58\)\( T^{2} + 23145039284 p^{13} T^{3} + p^{26} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 15623206500 T + \)\(18\!\cdots\!42\)\( T^{2} + 15623206500 p^{13} T^{3} + p^{26} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8132132168 T + \)\(23\!\cdots\!42\)\( T^{2} + 8132132168 p^{13} T^{3} + p^{26} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 124932588072 T + \)\(12\!\cdots\!50\)\( T^{2} - 124932588072 p^{13} T^{3} + p^{26} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 11471508804 T + \)\(38\!\cdots\!50\)\( T^{2} + 11471508804 p^{13} T^{3} + p^{26} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 569258151552 T + \)\(28\!\cdots\!34\)\( T^{2} - 569258151552 p^{13} T^{3} + p^{26} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 777684927236 T + \)\(37\!\cdots\!86\)\( T^{2} + 777684927236 p^{13} T^{3} + p^{26} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 65681674168 T + \)\(10\!\cdots\!90\)\( p T^{2} - 65681674168 p^{13} T^{3} + p^{26} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1276561230720 T + \)\(27\!\cdots\!22\)\( T^{2} - 1276561230720 p^{13} T^{3} + p^{26} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2262215682284 T + \)\(37\!\cdots\!30\)\( T^{2} + 2262215682284 p^{13} T^{3} + p^{26} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 130369818584 T + \)\(64\!\cdots\!42\)\( T^{2} + 130369818584 p^{13} T^{3} + p^{26} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1015329428952 T + \)\(77\!\cdots\!02\)\( T^{2} + 1015329428952 p^{13} T^{3} + p^{26} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 7032267407820 T + \)\(29\!\cdots\!38\)\( T^{2} - 7032267407820 p^{13} T^{3} + p^{26} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3304213236476 T + \)\(12\!\cdots\!98\)\( T^{2} + 3304213236476 p^{13} T^{3} + p^{26} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.48157631769447731368688450562, −10.33618906398960594491097110712, −9.236545083368094674783719332927, −9.226401841163119543033302073354, −8.607756375006619536988639998782, −8.476603018691410032759373701778, −7.20478044558405624355809102860, −7.03578067625053707556509122703, −6.36274631622960378031605676434, −6.31835552498527583909039746988, −5.39267492488116888613896246882, −5.02021137606662118515711142525, −4.32983179418509663015724186009, −3.89956353639767512570495127778, −2.96617825920659041556987594756, −2.76711034510193398971210359872, −1.93096822187202734946341663689, −1.79969354869669864464551168987, −0.74294456672507413306069703431, −0.32529673275556803627638372409,
0.32529673275556803627638372409, 0.74294456672507413306069703431, 1.79969354869669864464551168987, 1.93096822187202734946341663689, 2.76711034510193398971210359872, 2.96617825920659041556987594756, 3.89956353639767512570495127778, 4.32983179418509663015724186009, 5.02021137606662118515711142525, 5.39267492488116888613896246882, 6.31835552498527583909039746988, 6.36274631622960378031605676434, 7.03578067625053707556509122703, 7.20478044558405624355809102860, 8.476603018691410032759373701778, 8.607756375006619536988639998782, 9.226401841163119543033302073354, 9.236545083368094674783719332927, 10.33618906398960594491097110712, 10.48157631769447731368688450562
