| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 2·9-s + 14-s + 16-s + 3·17-s + 2·18-s + 9·23-s − 25-s + 28-s + 32-s + 3·34-s + 2·36-s − 3·41-s + 9·46-s − 12·47-s − 6·49-s − 50-s + 56-s + 2·63-s + 64-s + 3·68-s + 9·71-s + 2·72-s + 9·73-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 2/3·9-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.471·18-s + 1.87·23-s − 1/5·25-s + 0.188·28-s + 0.176·32-s + 0.514·34-s + 1/3·36-s − 0.468·41-s + 1.32·46-s − 1.75·47-s − 6/7·49-s − 0.141·50-s + 0.133·56-s + 0.251·63-s + 1/8·64-s + 0.363·68-s + 1.06·71-s + 0.235·72-s + 1.05·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.001952559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.001952559\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.369741822142111846136099199793, −8.720005054664145448969887033337, −8.177826745570557538704907472983, −7.76944629182084405721844892649, −7.25870392943920970674182113545, −6.59057460290633943192451225759, −6.54836688164540199503701943799, −5.49791478224113620916615669561, −5.23164870424698922283802066166, −4.70159077081551748031181823455, −4.12421419119836467199993872110, −3.38041189507005508798737873132, −2.95223702574570941693060489476, −1.94340548227415674373630939018, −1.17614541726880996836136951476,
1.17614541726880996836136951476, 1.94340548227415674373630939018, 2.95223702574570941693060489476, 3.38041189507005508798737873132, 4.12421419119836467199993872110, 4.70159077081551748031181823455, 5.23164870424698922283802066166, 5.49791478224113620916615669561, 6.54836688164540199503701943799, 6.59057460290633943192451225759, 7.25870392943920970674182113545, 7.76944629182084405721844892649, 8.177826745570557538704907472983, 8.720005054664145448969887033337, 9.369741822142111846136099199793
