| L(s) = 1 | + 0.954·2-s − 2.71·3-s − 1.08·4-s − 0.444·5-s − 2.59·6-s − 1.03·7-s − 2.94·8-s + 4.38·9-s − 0.423·10-s − 11-s + 2.96·12-s + 5.58·13-s − 0.987·14-s + 1.20·15-s − 0.634·16-s − 5.00·17-s + 4.18·18-s − 4.02·19-s + 0.483·20-s + 2.81·21-s − 0.954·22-s − 2.30·23-s + 8.01·24-s − 4.80·25-s + 5.32·26-s − 3.76·27-s + 1.12·28-s + ⋯ |
| L(s) = 1 | + 0.674·2-s − 1.56·3-s − 0.544·4-s − 0.198·5-s − 1.05·6-s − 0.391·7-s − 1.04·8-s + 1.46·9-s − 0.134·10-s − 0.301·11-s + 0.854·12-s + 1.54·13-s − 0.264·14-s + 0.311·15-s − 0.158·16-s − 1.21·17-s + 0.986·18-s − 0.922·19-s + 0.108·20-s + 0.613·21-s − 0.203·22-s − 0.481·23-s + 1.63·24-s − 0.960·25-s + 1.04·26-s − 0.724·27-s + 0.213·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 0.954T + 2T^{2} \) |
| 3 | \( 1 + 2.71T + 3T^{2} \) |
| 5 | \( 1 + 0.444T + 5T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 13 | \( 1 - 5.58T + 13T^{2} \) |
| 17 | \( 1 + 5.00T + 17T^{2} \) |
| 19 | \( 1 + 4.02T + 19T^{2} \) |
| 23 | \( 1 + 2.30T + 23T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 + 6.95T + 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 - 4.18T + 43T^{2} \) |
| 47 | \( 1 - 1.62T + 47T^{2} \) |
| 53 | \( 1 - 7.18T + 53T^{2} \) |
| 59 | \( 1 - 9.18T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 0.918T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 0.0583T + 79T^{2} \) |
| 83 | \( 1 - 6.59T + 83T^{2} \) |
| 89 | \( 1 - 3.12T + 89T^{2} \) |
| 97 | \( 1 + 4.98T + 97T^{2} \) |
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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
−6.84567766759766414848969793867, −6.50861180282304639320262164711, −5.89049555400210901421989178162, −5.42692731469709172675897674621, −4.52962919867482606127458114720, −4.14944292082775290407019792854, −3.40859599539704765005972228010, −2.19009591559844609997865785323, −0.848013880682105365780874777792, 0,
0.848013880682105365780874777792, 2.19009591559844609997865785323, 3.40859599539704765005972228010, 4.14944292082775290407019792854, 4.52962919867482606127458114720, 5.42692731469709172675897674621, 5.89049555400210901421989178162, 6.50861180282304639320262164711, 6.84567766759766414848969793867
