L(s) = 1 | + 2-s − 3-s − 4-s − 2·5-s − 6-s − 4·7-s − 3·8-s + 9-s − 2·10-s + 12-s − 4·14-s + 2·15-s − 16-s − 2·17-s + 18-s + 2·20-s + 4·21-s + 3·24-s − 25-s − 27-s + 4·28-s + 10·29-s + 2·30-s − 4·31-s + 5·32-s − 2·34-s + 8·35-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s − 1.51·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.288·12-s − 1.06·14-s + 0.516·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.447·20-s + 0.872·21-s + 0.612·24-s − 1/5·25-s − 0.192·27-s + 0.755·28-s + 1.85·29-s + 0.365·30-s − 0.718·31-s + 0.883·32-s − 0.342·34-s + 1.35·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7648565900\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7648565900\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.16745000262382, −13.71020517727744, −13.21805781294176, −12.64889679757722, −12.37480056523753, −12.05560593379512, −11.30436378240334, −10.89200086855354, −10.14591472748444, −9.762800571679534, −9.102113105066873, −8.796334824137330, −7.976579404437257, −7.443549080533481, −6.765741370983847, −6.277477031219613, −5.884272945129033, −5.227966145158816, −4.493232898747585, −4.112821687997470, −3.625184673458952, −2.949382040913699, −2.421055503432827, −0.9715959344555765, −0.3439385846995256,
0.3439385846995256, 0.9715959344555765, 2.421055503432827, 2.949382040913699, 3.625184673458952, 4.112821687997470, 4.493232898747585, 5.227966145158816, 5.884272945129033, 6.277477031219613, 6.765741370983847, 7.443549080533481, 7.976579404437257, 8.796334824137330, 9.102113105066873, 9.762800571679534, 10.14591472748444, 10.89200086855354, 11.30436378240334, 12.05560593379512, 12.37480056523753, 12.64889679757722, 13.21805781294176, 13.71020517727744, 14.16745000262382