L(s) = 1 | − 2-s + 4-s − 4·5-s − 7-s − 8-s − 3·9-s + 4·10-s − 11-s + 2·13-s + 14-s + 16-s − 4·17-s + 3·18-s − 6·19-s − 4·20-s + 22-s + 4·23-s + 11·25-s − 2·26-s − 28-s − 2·29-s − 2·31-s − 32-s + 4·34-s + 4·35-s − 3·36-s + 10·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.377·7-s − 0.353·8-s − 9-s + 1.26·10-s − 0.301·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.707·18-s − 1.37·19-s − 0.894·20-s + 0.213·22-s + 0.834·23-s + 11/5·25-s − 0.392·26-s − 0.188·28-s − 0.371·29-s − 0.359·31-s − 0.176·32-s + 0.685·34-s + 0.676·35-s − 1/2·36-s + 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.22063198281703280312046818474, −11.19149675055500729441636567526, −10.82425456930030561555637854181, −9.025591496790457491942383223256, −8.383894826086909083235343969228, −7.41620352139467519733544989343, −6.22872197038012242892904428945, −4.35107164128550817055960303512, −2.98342596760904293099474363551, 0,
2.98342596760904293099474363551, 4.35107164128550817055960303512, 6.22872197038012242892904428945, 7.41620352139467519733544989343, 8.383894826086909083235343969228, 9.025591496790457491942383223256, 10.82425456930030561555637854181, 11.19149675055500729441636567526, 12.22063198281703280312046818474