L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 11-s − 2·13-s + 14-s + 16-s − 6·17-s + 8·19-s + 20-s + 22-s + 25-s + 2·26-s − 28-s + 2·29-s − 4·31-s − 32-s + 6·34-s − 35-s − 2·37-s − 8·38-s − 40-s + 2·41-s + 4·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 1.83·19-s + 0.223·20-s + 0.213·22-s + 1/5·25-s + 0.392·26-s − 0.188·28-s + 0.371·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s − 0.169·35-s − 0.328·37-s − 1.29·38-s − 0.158·40-s + 0.312·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.53344017742354297144926711277, −7.03972240918578718273997613566, −6.36560780417769818387248969571, −5.53277858145217564714655351081, −4.92613865022551413118169164212, −3.86729109294964278379975262687, −2.89062496400152127399999429528, −2.27394211790160698973340505992, −1.20670669357269219384399860358, 0,
1.20670669357269219384399860358, 2.27394211790160698973340505992, 2.89062496400152127399999429528, 3.86729109294964278379975262687, 4.92613865022551413118169164212, 5.53277858145217564714655351081, 6.36560780417769818387248969571, 7.03972240918578718273997613566, 7.53344017742354297144926711277