L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s − 3·9-s − 11-s − 6·13-s + 14-s − 16-s + 2·17-s − 3·18-s − 8·19-s − 22-s − 6·26-s − 28-s + 29-s − 4·31-s + 5·32-s + 2·34-s + 3·36-s + 2·37-s − 8·38-s − 2·41-s + 4·43-s + 44-s + 4·47-s + 49-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 9-s − 0.301·11-s − 1.66·13-s + 0.267·14-s − 1/4·16-s + 0.485·17-s − 0.707·18-s − 1.83·19-s − 0.213·22-s − 1.17·26-s − 0.188·28-s + 0.185·29-s − 0.718·31-s + 0.883·32-s + 0.342·34-s + 1/2·36-s + 0.328·37-s − 1.29·38-s − 0.312·41-s + 0.609·43-s + 0.150·44-s + 0.583·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.67409345828806, −14.21850912577226, −13.69858052810866, −13.11007591339277, −12.62963029873533, −12.19469659053903, −11.82892276704355, −11.12938698152169, −10.61146169478727, −10.01033788886852, −9.508767686247038, −8.838229638494224, −8.480458368830804, −7.952507892943105, −7.262625528225798, −6.680332375371514, −5.888857667085541, −5.523897510456000, −5.011508643020376, −4.428656729289803, −3.936707874163119, −3.163064562602785, −2.448354476687644, −2.140574473020192, −0.7069924217443032, 0,
0.7069924217443032, 2.140574473020192, 2.448354476687644, 3.163064562602785, 3.936707874163119, 4.428656729289803, 5.011508643020376, 5.523897510456000, 5.888857667085541, 6.680332375371514, 7.262625528225798, 7.952507892943105, 8.480458368830804, 8.838229638494224, 9.508767686247038, 10.01033788886852, 10.61146169478727, 11.12938698152169, 11.82892276704355, 12.19469659053903, 12.62963029873533, 13.11007591339277, 13.69858052810866, 14.21850912577226, 14.67409345828806