L(s) = 1 | − 14·5-s + 24·7-s − 28·11-s − 74·13-s − 82·17-s − 92·19-s + 8·23-s + 71·25-s + 138·29-s − 80·31-s − 336·35-s + 30·37-s − 282·41-s − 4·43-s + 240·47-s + 233·49-s + 130·53-s + 392·55-s + 596·59-s − 218·61-s + 1.03e3·65-s + 436·67-s + 856·71-s − 998·73-s − 672·77-s + 32·79-s − 1.50e3·83-s + ⋯ |
L(s) = 1 | − 1.25·5-s + 1.29·7-s − 0.767·11-s − 1.57·13-s − 1.16·17-s − 1.11·19-s + 0.0725·23-s + 0.567·25-s + 0.883·29-s − 0.463·31-s − 1.62·35-s + 0.133·37-s − 1.07·41-s − 0.0141·43-s + 0.744·47-s + 0.679·49-s + 0.336·53-s + 0.961·55-s + 1.31·59-s − 0.457·61-s + 1.97·65-s + 0.795·67-s + 1.43·71-s − 1.60·73-s − 0.994·77-s + 0.0455·79-s − 1.99·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 14 T + p^{3} T^{2} \) |
| 7 | \( 1 - 24 T + p^{3} T^{2} \) |
| 11 | \( 1 + 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 74 T + p^{3} T^{2} \) |
| 17 | \( 1 + 82 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 8 T + p^{3} T^{2} \) |
| 29 | \( 1 - 138 T + p^{3} T^{2} \) |
| 31 | \( 1 + 80 T + p^{3} T^{2} \) |
| 37 | \( 1 - 30 T + p^{3} T^{2} \) |
| 41 | \( 1 + 282 T + p^{3} T^{2} \) |
| 43 | \( 1 + 4 T + p^{3} T^{2} \) |
| 47 | \( 1 - 240 T + p^{3} T^{2} \) |
| 53 | \( 1 - 130 T + p^{3} T^{2} \) |
| 59 | \( 1 - 596 T + p^{3} T^{2} \) |
| 61 | \( 1 + 218 T + p^{3} T^{2} \) |
| 67 | \( 1 - 436 T + p^{3} T^{2} \) |
| 71 | \( 1 - 856 T + p^{3} T^{2} \) |
| 73 | \( 1 + 998 T + p^{3} T^{2} \) |
| 79 | \( 1 - 32 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1508 T + p^{3} T^{2} \) |
| 89 | \( 1 - 246 T + p^{3} T^{2} \) |
| 97 | \( 1 - 866 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.98210805944617396946787451006, −11.23685179299906162796153946451, −10.30457650302398902601452221009, −8.666587282283757441146121082801, −7.909222333074774902226423421001, −7.00326137857841809134295492785, −5.05396551381953905744109524746, −4.26526896445566079715371921762, −2.34570396205530952866409994002, 0,
2.34570396205530952866409994002, 4.26526896445566079715371921762, 5.05396551381953905744109524746, 7.00326137857841809134295492785, 7.909222333074774902226423421001, 8.666587282283757441146121082801, 10.30457650302398902601452221009, 11.23685179299906162796153946451, 11.98210805944617396946787451006