| L(s) = 1 | − 2.08·2-s − 3-s + 2.35·4-s − 5-s + 2.08·6-s + 5.08·7-s − 0.734·8-s + 9-s + 2.08·10-s − 4.17·11-s − 2.35·12-s − 1.08·13-s − 10.6·14-s + 15-s − 3.17·16-s + 0.648·17-s − 2.08·18-s − 2.70·19-s − 2.35·20-s − 5.08·21-s + 8.70·22-s + 7.52·23-s + 0.734·24-s + 25-s + 2.26·26-s − 27-s + 11.9·28-s + ⋯ |
| L(s) = 1 | − 1.47·2-s − 0.577·3-s + 1.17·4-s − 0.447·5-s + 0.851·6-s + 1.92·7-s − 0.259·8-s + 0.333·9-s + 0.659·10-s − 1.25·11-s − 0.678·12-s − 0.301·13-s − 2.83·14-s + 0.258·15-s − 0.793·16-s + 0.157·17-s − 0.491·18-s − 0.620·19-s − 0.525·20-s − 1.10·21-s + 1.85·22-s + 1.56·23-s + 0.149·24-s + 0.200·25-s + 0.444·26-s − 0.192·27-s + 2.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5809234310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5809234310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.08T + 2T^{2} \) |
| 7 | \( 1 - 5.08T + 7T^{2} \) |
| 11 | \( 1 + 4.17T + 11T^{2} \) |
| 13 | \( 1 + 1.08T + 13T^{2} \) |
| 17 | \( 1 - 0.648T + 17T^{2} \) |
| 19 | \( 1 + 2.70T + 19T^{2} \) |
| 23 | \( 1 - 7.52T + 23T^{2} \) |
| 29 | \( 1 - 5.90T + 29T^{2} \) |
| 37 | \( 1 - 0.913T + 37T^{2} \) |
| 41 | \( 1 - 2.17T + 41T^{2} \) |
| 43 | \( 1 - 8.17T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 5.52T + 53T^{2} \) |
| 59 | \( 1 - 0.438T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 9.79T + 67T^{2} \) |
| 71 | \( 1 - 2.96T + 71T^{2} \) |
| 73 | \( 1 + 1.25T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−10.86392227234339800324779354852, −10.38811766668446675338111057076, −9.122002582250336767344447483504, −8.146021261385395460694139023083, −7.81894606951453616006703951522, −6.84669169549494150000830406401, −5.20525631429268834504227468159, −4.57971232276313505694500890357, −2.34077005085669020253770399207, −0.937189638722921466255912033579,
0.937189638722921466255912033579, 2.34077005085669020253770399207, 4.57971232276313505694500890357, 5.20525631429268834504227468159, 6.84669169549494150000830406401, 7.81894606951453616006703951522, 8.146021261385395460694139023083, 9.122002582250336767344447483504, 10.38811766668446675338111057076, 10.86392227234339800324779354852
