L(s) = 1 | − 2-s + 3-s − 4-s − 6-s − 7-s + 3·8-s + 9-s − 11-s − 12-s + 2·13-s + 14-s − 16-s + 6·17-s − 18-s + 4·19-s − 21-s + 22-s + 4·23-s + 3·24-s − 2·26-s + 27-s + 28-s − 2·29-s − 4·31-s − 5·32-s − 33-s − 6·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.554·13-s + 0.267·14-s − 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.218·21-s + 0.213·22-s + 0.834·23-s + 0.612·24-s − 0.392·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s − 0.718·31-s − 0.883·32-s − 0.174·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.513007850\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.513007850\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.068342640019531914978250428128, −7.62473931438603580998033976144, −7.07479461883781106094593753786, −5.88414851043740476269231571092, −5.29918809235745225266491193655, −4.35859904059561991914721462173, −3.53745608067103991720675741680, −2.89791543143922995015902880894, −1.58932132421866959375513549390, −0.76396374833741058383114104263,
0.76396374833741058383114104263, 1.58932132421866959375513549390, 2.89791543143922995015902880894, 3.53745608067103991720675741680, 4.35859904059561991914721462173, 5.29918809235745225266491193655, 5.88414851043740476269231571092, 7.07479461883781106094593753786, 7.62473931438603580998033976144, 8.068342640019531914978250428128