| L(s) = 1 | + 1.59·3-s + 3.07·5-s − 0.447·9-s − 4.02·11-s − 0.691·13-s + 4.90·15-s − 1.37·17-s − 0.475·19-s + 23-s + 4.43·25-s − 5.50·27-s − 4.32·29-s + 2.13·31-s − 6.42·33-s − 7.96·37-s − 1.10·39-s − 6.24·41-s − 8.22·43-s − 1.37·45-s + 2.96·47-s − 2.19·51-s − 11.0·53-s − 12.3·55-s − 0.759·57-s + 1.89·59-s + 11.5·61-s − 2.12·65-s + ⋯ |
| L(s) = 1 | + 0.922·3-s + 1.37·5-s − 0.149·9-s − 1.21·11-s − 0.191·13-s + 1.26·15-s − 0.332·17-s − 0.109·19-s + 0.208·23-s + 0.886·25-s − 1.05·27-s − 0.802·29-s + 0.383·31-s − 1.11·33-s − 1.31·37-s − 0.177·39-s − 0.975·41-s − 1.25·43-s − 0.204·45-s + 0.432·47-s − 0.306·51-s − 1.52·53-s − 1.66·55-s − 0.100·57-s + 0.246·59-s + 1.47·61-s − 0.263·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 1.59T + 3T^{2} \) |
| 5 | \( 1 - 3.07T + 5T^{2} \) |
| 11 | \( 1 + 4.02T + 11T^{2} \) |
| 13 | \( 1 + 0.691T + 13T^{2} \) |
| 17 | \( 1 + 1.37T + 17T^{2} \) |
| 19 | \( 1 + 0.475T + 19T^{2} \) |
| 29 | \( 1 + 4.32T + 29T^{2} \) |
| 31 | \( 1 - 2.13T + 31T^{2} \) |
| 37 | \( 1 + 7.96T + 37T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 + 8.22T + 43T^{2} \) |
| 47 | \( 1 - 2.96T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 1.89T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 1.64T + 67T^{2} \) |
| 71 | \( 1 + 9.61T + 71T^{2} \) |
| 73 | \( 1 - 2.60T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 - 5.38T + 83T^{2} \) |
| 89 | \( 1 + 1.41T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−7.46135992685063551287280758179, −6.74399926520181237290172385895, −5.93882250503069313918168354510, −5.34752388186602683612655308170, −4.78304674597125397177804768609, −3.57671618858078426409058605456, −2.90832159369933954477016163469, −2.21695126986043585314749643439, −1.66330914789271331797105422483, 0,
1.66330914789271331797105422483, 2.21695126986043585314749643439, 2.90832159369933954477016163469, 3.57671618858078426409058605456, 4.78304674597125397177804768609, 5.34752388186602683612655308170, 5.93882250503069313918168354510, 6.74399926520181237290172385895, 7.46135992685063551287280758179
