L(s) = 1 | + 2·2-s + 4·3-s − 28·4-s − 25·5-s + 8·6-s − 120·8-s − 227·9-s − 50·10-s − 148·11-s − 112·12-s − 286·13-s − 100·15-s + 656·16-s + 1.67e3·17-s − 454·18-s − 1.06e3·19-s + 700·20-s − 296·22-s + 2.97e3·23-s − 480·24-s + 625·25-s − 572·26-s − 1.88e3·27-s − 3.41e3·29-s − 200·30-s + 2.44e3·31-s + 5.15e3·32-s + ⋯ |
L(s) = 1 | + 0.353·2-s + 0.256·3-s − 7/8·4-s − 0.447·5-s + 0.0907·6-s − 0.662·8-s − 0.934·9-s − 0.158·10-s − 0.368·11-s − 0.224·12-s − 0.469·13-s − 0.114·15-s + 0.640·16-s + 1.40·17-s − 0.330·18-s − 0.673·19-s + 0.391·20-s − 0.130·22-s + 1.17·23-s − 0.170·24-s + 1/5·25-s − 0.165·26-s − 0.496·27-s − 0.752·29-s − 0.0405·30-s + 0.457·31-s + 0.889·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.362191020\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.362191020\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + p^{2} T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - p T + p^{5} T^{2} \) |
| 3 | \( 1 - 4 T + p^{5} T^{2} \) |
| 11 | \( 1 + 148 T + p^{5} T^{2} \) |
| 13 | \( 1 + 22 p T + p^{5} T^{2} \) |
| 17 | \( 1 - 1678 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1060 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2976 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3410 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2448 T + p^{5} T^{2} \) |
| 37 | \( 1 - 182 T + p^{5} T^{2} \) |
| 41 | \( 1 - 9398 T + p^{5} T^{2} \) |
| 43 | \( 1 + 1244 T + p^{5} T^{2} \) |
| 47 | \( 1 - 12088 T + p^{5} T^{2} \) |
| 53 | \( 1 - 23846 T + p^{5} T^{2} \) |
| 59 | \( 1 - 20020 T + p^{5} T^{2} \) |
| 61 | \( 1 + 32302 T + p^{5} T^{2} \) |
| 67 | \( 1 - 60972 T + p^{5} T^{2} \) |
| 71 | \( 1 + 32648 T + p^{5} T^{2} \) |
| 73 | \( 1 - 38774 T + p^{5} T^{2} \) |
| 79 | \( 1 + 33360 T + p^{5} T^{2} \) |
| 83 | \( 1 + 16716 T + p^{5} T^{2} \) |
| 89 | \( 1 + 101370 T + p^{5} T^{2} \) |
| 97 | \( 1 - 119038 T + p^{5} T^{2} \) |
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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
−11.36486056211701427070013667621, −10.20391635548427364711426688711, −9.163127495753059572419585194562, −8.365629225637249283591375542418, −7.44717982173555021370698562607, −5.85071453229782270722461301097, −5.00267729164143909410617395082, −3.76579732720822462358053902204, −2.75311277071952905253925310886, −0.63450262739949142060609660852,
0.63450262739949142060609660852, 2.75311277071952905253925310886, 3.76579732720822462358053902204, 5.00267729164143909410617395082, 5.85071453229782270722461301097, 7.44717982173555021370698562607, 8.365629225637249283591375542418, 9.163127495753059572419585194562, 10.20391635548427364711426688711, 11.36486056211701427070013667621