| L(s) = 1 | − 5-s − 9-s − 13-s − 4·17-s − 4·25-s − 29-s + 9·37-s − 2·41-s + 45-s + 11·49-s − 6·53-s + 23·61-s + 65-s + 8·73-s + 81-s + 4·85-s − 23·89-s − 8·97-s + 3·101-s − 12·109-s − 22·113-s + 117-s − 4·121-s + 9·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1/3·9-s − 0.277·13-s − 0.970·17-s − 4/5·25-s − 0.185·29-s + 1.47·37-s − 0.312·41-s + 0.149·45-s + 11/7·49-s − 0.824·53-s + 2.94·61-s + 0.124·65-s + 0.936·73-s + 1/9·81-s + 0.433·85-s − 2.43·89-s − 0.812·97-s + 0.298·101-s − 1.14·109-s − 2.06·113-s + 0.0924·117-s − 0.363·121-s + 0.804·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.199318410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.199318410\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 93 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.095331438467123727469179989924, −7.88780105067965570049654899640, −7.19497412193321552385471907583, −6.93282436133300946919588774849, −6.44679879759132883129171173718, −5.92663136663015042317629187401, −5.44026769337605241088335239807, −5.06415227877115303842649043378, −4.32534889813636400909263180206, −4.03608553425935888132377002087, −3.58593597371221436366052457333, −2.60912326806563520733484157785, −2.49880612339475518951382066339, −1.55630994559044765891072901514, −0.51223893855862101040116561387,
0.51223893855862101040116561387, 1.55630994559044765891072901514, 2.49880612339475518951382066339, 2.60912326806563520733484157785, 3.58593597371221436366052457333, 4.03608553425935888132377002087, 4.32534889813636400909263180206, 5.06415227877115303842649043378, 5.44026769337605241088335239807, 5.92663136663015042317629187401, 6.44679879759132883129171173718, 6.93282436133300946919588774849, 7.19497412193321552385471907583, 7.88780105067965570049654899640, 8.095331438467123727469179989924
