| L(s) = 1 | − 5.36·2-s + 3·3-s + 20.7·4-s + 2.69·5-s − 16.0·6-s + 15.2·7-s − 68.5·8-s + 9·9-s − 14.4·10-s − 66.8·11-s + 62.3·12-s − 81.5·14-s + 8.08·15-s + 201.·16-s + 4.16·17-s − 48.2·18-s + 26.0·19-s + 56.0·20-s + 45.6·21-s + 358.·22-s + 47.3·23-s − 205.·24-s − 117.·25-s + 27·27-s + 315.·28-s + 257.·29-s − 43.3·30-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 0.577·3-s + 2.59·4-s + 0.241·5-s − 1.09·6-s + 0.820·7-s − 3.03·8-s + 0.333·9-s − 0.457·10-s − 1.83·11-s + 1.49·12-s − 1.55·14-s + 0.139·15-s + 3.14·16-s + 0.0594·17-s − 0.632·18-s + 0.314·19-s + 0.626·20-s + 0.473·21-s + 3.47·22-s + 0.429·23-s − 1.74·24-s − 0.941·25-s + 0.192·27-s + 2.13·28-s + 1.64·29-s − 0.264·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.036958037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.036958037\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 5.36T + 8T^{2} \) |
| 5 | \( 1 - 2.69T + 125T^{2} \) |
| 7 | \( 1 - 15.2T + 343T^{2} \) |
| 11 | \( 1 + 66.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 4.16T + 4.91e3T^{2} \) |
| 19 | \( 1 - 26.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 47.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 257.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 206.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 175.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 156.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 51.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 354.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 10.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 445.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 119.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 22.4T + 3.00e5T^{2} \) |
| 71 | \( 1 - 285.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 740.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 547.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 603.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 215.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.44e3T + 9.12e5T^{2} \) |
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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
−10.18626469744715005230456944119, −9.697453351880264943999908521871, −8.508602949873867661428037617495, −8.057917550880501202535927941494, −7.44918223507081942194450507813, −6.26489176961936937539319986463, −4.96627401109933075391853403715, −2.90055856612341500909020224567, −2.12236369668891410282893892752, −0.808707953307720000271987847266,
0.808707953307720000271987847266, 2.12236369668891410282893892752, 2.90055856612341500909020224567, 4.96627401109933075391853403715, 6.26489176961936937539319986463, 7.44918223507081942194450507813, 8.057917550880501202535927941494, 8.508602949873867661428037617495, 9.697453351880264943999908521871, 10.18626469744715005230456944119
