| L(s) = 1 | − 0.954·2-s − 0.726·3-s − 1.08·4-s − 5-s + 0.693·6-s − 1.92·7-s + 2.94·8-s − 2.47·9-s + 0.954·10-s − 2.82·11-s + 0.791·12-s − 6.44·13-s + 1.84·14-s + 0.726·15-s − 0.634·16-s − 2.15·17-s + 2.35·18-s + 1.30·19-s + 1.08·20-s + 1.40·21-s + 2.69·22-s + 1.78·23-s − 2.14·24-s + 25-s + 6.15·26-s + 3.97·27-s + 2.10·28-s + ⋯ |
| L(s) = 1 | − 0.674·2-s − 0.419·3-s − 0.544·4-s − 0.447·5-s + 0.283·6-s − 0.729·7-s + 1.04·8-s − 0.824·9-s + 0.301·10-s − 0.852·11-s + 0.228·12-s − 1.78·13-s + 0.492·14-s + 0.187·15-s − 0.158·16-s − 0.523·17-s + 0.556·18-s + 0.299·19-s + 0.243·20-s + 0.305·21-s + 0.575·22-s + 0.372·23-s − 0.437·24-s + 0.200·25-s + 1.20·26-s + 0.765·27-s + 0.397·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2105567805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2105567805\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 0.954T + 2T^{2} \) |
| 3 | \( 1 + 0.726T + 3T^{2} \) |
| 7 | \( 1 + 1.92T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 6.44T + 13T^{2} \) |
| 17 | \( 1 + 2.15T + 17T^{2} \) |
| 19 | \( 1 - 1.30T + 19T^{2} \) |
| 23 | \( 1 - 1.78T + 23T^{2} \) |
| 29 | \( 1 - 3.74T + 29T^{2} \) |
| 31 | \( 1 + 8.69T + 31T^{2} \) |
| 37 | \( 1 + 5.53T + 37T^{2} \) |
| 41 | \( 1 - 2.83T + 41T^{2} \) |
| 43 | \( 1 - 4.52T + 43T^{2} \) |
| 47 | \( 1 + 9.40T + 47T^{2} \) |
| 53 | \( 1 - 4.21T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 9.42T + 71T^{2} \) |
| 73 | \( 1 - 16.8T + 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 - 5.44T + 83T^{2} \) |
| 89 | \( 1 - 6.41T + 89T^{2} \) |
| 97 | \( 1 - 3.18T + 97T^{2} \) |
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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
−9.611881001184278303782952740455, −9.081357352125506595836912279420, −8.107688649478178149969166766182, −7.46460729541397509143767096083, −6.59487521265867801918258798115, −5.26100946225511189970412643138, −4.87386583631422706792352013400, −3.51187207079476971994281417762, −2.39497746068013647477970071915, −0.36325618024072769996679555773,
0.36325618024072769996679555773, 2.39497746068013647477970071915, 3.51187207079476971994281417762, 4.87386583631422706792352013400, 5.26100946225511189970412643138, 6.59487521265867801918258798115, 7.46460729541397509143767096083, 8.107688649478178149969166766182, 9.081357352125506595836912279420, 9.611881001184278303782952740455
