| L(s) = 1 | − 7·2-s + 25·4-s + 25·7-s − 63·8-s + 22·11-s + 50·13-s − 175·14-s + 169·16-s − 151·17-s − 3·19-s − 154·22-s + 48·23-s − 350·26-s + 625·28-s + 221·29-s + 141·31-s − 623·32-s + 1.05e3·34-s + 559·37-s + 21·38-s + 144·41-s + 60·43-s + 550·44-s − 336·46-s − 48·47-s + 127·49-s + 1.25e3·52-s + ⋯ |
| L(s) = 1 | − 2.47·2-s + 25/8·4-s + 1.34·7-s − 2.78·8-s + 0.603·11-s + 1.06·13-s − 3.34·14-s + 2.64·16-s − 2.15·17-s − 0.0362·19-s − 1.49·22-s + 0.435·23-s − 2.64·26-s + 4.21·28-s + 1.41·29-s + 0.816·31-s − 3.44·32-s + 5.33·34-s + 2.48·37-s + 0.0896·38-s + 0.548·41-s + 0.212·43-s + 1.88·44-s − 1.07·46-s − 0.148·47-s + 0.370·49-s + 3.33·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9490975131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9490975131\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 7 T + 3 p^{3} T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 25 T + 498 T^{2} - 25 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 50 T + 4594 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 151 T + 14298 T^{2} + 151 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 5114 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 48 T + 24842 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 221 T + 39564 T^{2} - 221 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 141 T + 6374 T^{2} - 141 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 559 T + 171568 T^{2} - 559 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 144 T + 39598 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 60 T + 56486 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 48 T + 3526 p T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 117 T + 179792 T^{2} - 117 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 86 T + 306510 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 155 T + 399784 T^{2} + 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 266 T + 523590 T^{2} + 266 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1587 T + 1329650 T^{2} + 1587 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 70 T + 764962 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1294 T + 1349454 T^{2} + 1294 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 558 T + 911590 T^{2} + 558 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1777 T + 2191512 T^{2} - 1777 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 334 T + 977242 T^{2} - 334 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.705511863156371384190229700646, −8.650642842885239865267646258897, −8.108192799794567087244957511553, −7.87885320910221136765631343248, −7.47217664194056023401289043326, −7.13581827526283837654995668023, −6.55434574309393826552331503202, −6.19300133631739342570347466613, −6.07320612588262219219533089877, −5.35671814215481847010813043386, −4.60383467572379345492065702527, −4.50194385270969229534979394340, −4.06776977973669095190490156770, −3.29025544063504161326413377142, −2.64040219117758273629868311359, −2.35675632314704089251216080042, −1.56671103981907851463601270980, −1.34262396739187894758984878955, −0.887061440225823710736722321107, −0.37525469839682219589408221175,
0.37525469839682219589408221175, 0.887061440225823710736722321107, 1.34262396739187894758984878955, 1.56671103981907851463601270980, 2.35675632314704089251216080042, 2.64040219117758273629868311359, 3.29025544063504161326413377142, 4.06776977973669095190490156770, 4.50194385270969229534979394340, 4.60383467572379345492065702527, 5.35671814215481847010813043386, 6.07320612588262219219533089877, 6.19300133631739342570347466613, 6.55434574309393826552331503202, 7.13581827526283837654995668023, 7.47217664194056023401289043326, 7.87885320910221136765631343248, 8.108192799794567087244957511553, 8.650642842885239865267646258897, 8.705511863156371384190229700646
