| L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s − 3·8-s − 2·9-s + 10-s − 12-s + 15-s − 16-s − 2·18-s − 20-s − 3·24-s + 25-s − 5·27-s + 30-s + 7·31-s + 5·32-s + 2·36-s + 9·37-s − 3·40-s + 3·41-s + 6·43-s − 2·45-s − 48-s + 11·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.06·8-s − 2/3·9-s + 0.316·10-s − 0.288·12-s + 0.258·15-s − 1/4·16-s − 0.471·18-s − 0.223·20-s − 0.612·24-s + 1/5·25-s − 0.962·27-s + 0.182·30-s + 1.25·31-s + 0.883·32-s + 1/3·36-s + 1.47·37-s − 0.474·40-s + 0.468·41-s + 0.914·43-s − 0.298·45-s − 0.144·48-s + 11/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.372686542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.372686542\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
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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
−9.041495130659913291796609340518, −8.619379858860742014519478097760, −8.115254595515861403192492614635, −7.82067700085579504791403153644, −7.05437394898644771902617788780, −6.50305877557587626274725562445, −5.90107096445752491992628901869, −5.67717689057213501223549252868, −5.08566268154462671636883732769, −4.34086900423517697146667239338, −4.09772230576046651221897837482, −3.24142362144906189270559175901, −2.75913019632310416779262742028, −2.22449489479160615519463093887, −0.851968183094800304293292875653,
0.851968183094800304293292875653, 2.22449489479160615519463093887, 2.75913019632310416779262742028, 3.24142362144906189270559175901, 4.09772230576046651221897837482, 4.34086900423517697146667239338, 5.08566268154462671636883732769, 5.67717689057213501223549252868, 5.90107096445752491992628901869, 6.50305877557587626274725562445, 7.05437394898644771902617788780, 7.82067700085579504791403153644, 8.115254595515861403192492614635, 8.619379858860742014519478097760, 9.041495130659913291796609340518
