| L(s) = 1 | − 3.12e4·5-s − 4.56e5·7-s − 8.50e6·11-s − 2.65e7·13-s + 1.56e8·17-s + 2.38e8·19-s + 1.39e9·23-s + 7.32e8·25-s + 8.66e9·29-s − 4.03e9·31-s + 1.42e10·35-s − 1.55e10·37-s − 2.16e10·41-s + 7.66e10·43-s + 1.06e10·47-s + 8.07e10·49-s + 1.66e11·53-s + 2.65e11·55-s − 1.83e11·59-s + 6.89e10·61-s + 8.30e11·65-s − 1.66e12·67-s + 2.99e11·71-s − 1.70e12·73-s + 3.87e12·77-s − 2.56e11·79-s + 4.27e12·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.46·7-s − 1.44·11-s − 1.52·13-s + 1.57·17-s + 1.16·19-s + 1.96·23-s + 3/5·25-s + 2.70·29-s − 0.816·31-s + 1.31·35-s − 0.994·37-s − 0.712·41-s + 1.84·43-s + 0.144·47-s + 0.833·49-s + 1.03·53-s + 1.29·55-s − 0.565·59-s + 0.171·61-s + 1.36·65-s − 2.25·67-s + 0.277·71-s − 1.31·73-s + 2.11·77-s − 0.118·79-s + 1.43·83-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
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
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 | $D_{4}$ | \( 1 + 456140 T + 18191521410 p T^{2} + 456140 p^{13} T^{3} + p^{26} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 8500080 T + 7114598702642 p T^{2} + 8500080 p^{13} T^{3} + p^{26} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 26573780 T + 708499347199902 T^{2} + 26573780 p^{13} T^{3} + p^{26} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 156978540 T + 21871312563328198 T^{2} - 156978540 p^{13} T^{3} + p^{26} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 238996312 T + 65310799772488854 T^{2} - 238996312 p^{13} T^{3} + p^{26} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1392511380 T + 981520481942884270 T^{2} - 1392511380 p^{13} T^{3} + p^{26} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8663268228 T + 35441505443412290974 T^{2} - 8663268228 p^{13} T^{3} + p^{26} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4033525448 T + 49011907812955175358 T^{2} + 4033525448 p^{13} T^{3} + p^{26} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 15514134260 T + \)\(15\!\cdots\!10\)\( T^{2} + 15514134260 p^{13} T^{3} + p^{26} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 21661691172 T + \)\(38\!\cdots\!38\)\( T^{2} + 21661691172 p^{13} T^{3} + p^{26} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 76678922260 T + \)\(47\!\cdots\!86\)\( T^{2} - 76678922260 p^{13} T^{3} + p^{26} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10650423900 T + \)\(49\!\cdots\!78\)\( T^{2} - 10650423900 p^{13} T^{3} + p^{26} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 166929023460 T + \)\(58\!\cdots\!50\)\( T^{2} - 166929023460 p^{13} T^{3} + p^{26} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 183106613304 T + \)\(20\!\cdots\!62\)\( T^{2} + 183106613304 p^{13} T^{3} + p^{26} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 68904440284 T + \)\(40\!\cdots\!26\)\( T^{2} - 68904440284 p^{13} T^{3} + p^{26} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1668124615460 T + \)\(17\!\cdots\!70\)\( T^{2} + 1668124615460 p^{13} T^{3} + p^{26} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 299938755864 T - \)\(98\!\cdots\!54\)\( T^{2} - 299938755864 p^{13} T^{3} + p^{26} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1702692739580 T + \)\(35\!\cdots\!02\)\( T^{2} + 1702692739580 p^{13} T^{3} + p^{26} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 256669983344 T + \)\(90\!\cdots\!62\)\( T^{2} + 256669983344 p^{13} T^{3} + p^{26} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4277566394700 T + \)\(19\!\cdots\!30\)\( T^{2} - 4277566394700 p^{13} T^{3} + p^{26} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5848281889428 T + \)\(40\!\cdots\!34\)\( T^{2} + 5848281889428 p^{13} T^{3} + p^{26} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1479529172780 T + \)\(95\!\cdots\!10\)\( T^{2} + 1479529172780 p^{13} T^{3} + p^{26} T^{4} \) |
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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
−10.07171841756438310738707243742, −9.656281906262629746930123959161, −9.144673302111358573919895514323, −8.594011616358308598614013192453, −7.900597520278630683304827166108, −7.53952101369323871745709121612, −6.96131170680303205698437363029, −6.94943805493176022908894577730, −5.76991939225481545594730667649, −5.54431731237908082229859208876, −4.76036453051197845106000802872, −4.63882945571361479010627926107, −3.45910633125702664050787974724, −3.23062235277760525533920647221, −2.78703414362662348428446295690, −2.45427086896905320905418807989, −1.11519298936573385615986398463, −0.926340277646028531358621023490, 0, 0,
0.926340277646028531358621023490, 1.11519298936573385615986398463, 2.45427086896905320905418807989, 2.78703414362662348428446295690, 3.23062235277760525533920647221, 3.45910633125702664050787974724, 4.63882945571361479010627926107, 4.76036453051197845106000802872, 5.54431731237908082229859208876, 5.76991939225481545594730667649, 6.94943805493176022908894577730, 6.96131170680303205698437363029, 7.53952101369323871745709121612, 7.900597520278630683304827166108, 8.594011616358308598614013192453, 9.144673302111358573919895514323, 9.656281906262629746930123959161, 10.07171841756438310738707243742
