| L(s) = 1 | − 0.941·3-s − 2.86·5-s + 1.50·7-s − 2.11·9-s + 4.17·11-s + 2.74·13-s + 2.69·15-s − 0.166·17-s − 3.61·19-s − 1.41·21-s − 4.35·23-s + 3.18·25-s + 4.81·27-s + 0.193·29-s + 2.48·31-s − 3.93·33-s − 4.29·35-s + 1.39·37-s − 2.58·39-s − 3.49·41-s − 3.12·43-s + 6.04·45-s − 9.20·47-s − 4.74·49-s + 0.157·51-s − 6.24·53-s − 11.9·55-s + ⋯ |
| L(s) = 1 | − 0.543·3-s − 1.27·5-s + 0.567·7-s − 0.704·9-s + 1.25·11-s + 0.760·13-s + 0.695·15-s − 0.0404·17-s − 0.828·19-s − 0.308·21-s − 0.907·23-s + 0.636·25-s + 0.926·27-s + 0.0359·29-s + 0.446·31-s − 0.684·33-s − 0.726·35-s + 0.229·37-s − 0.413·39-s − 0.545·41-s − 0.476·43-s + 0.901·45-s − 1.34·47-s − 0.677·49-s + 0.0219·51-s − 0.858·53-s − 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 3 | \( 1 + 0.941T + 3T^{2} \) |
| 5 | \( 1 + 2.86T + 5T^{2} \) |
| 7 | \( 1 - 1.50T + 7T^{2} \) |
| 11 | \( 1 - 4.17T + 11T^{2} \) |
| 13 | \( 1 - 2.74T + 13T^{2} \) |
| 17 | \( 1 + 0.166T + 17T^{2} \) |
| 19 | \( 1 + 3.61T + 19T^{2} \) |
| 23 | \( 1 + 4.35T + 23T^{2} \) |
| 29 | \( 1 - 0.193T + 29T^{2} \) |
| 31 | \( 1 - 2.48T + 31T^{2} \) |
| 37 | \( 1 - 1.39T + 37T^{2} \) |
| 41 | \( 1 + 3.49T + 41T^{2} \) |
| 43 | \( 1 + 3.12T + 43T^{2} \) |
| 47 | \( 1 + 9.20T + 47T^{2} \) |
| 53 | \( 1 + 6.24T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 3.14T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 4.35T + 71T^{2} \) |
| 73 | \( 1 - 0.531T + 73T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 - 0.923T + 83T^{2} \) |
| 89 | \( 1 - 2.34T + 89T^{2} \) |
| 97 | \( 1 - 7.23T + 97T^{2} \) |
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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
−8.964905150568887964535101070416, −8.372034008912819522598019778511, −7.74963319311631313863943114023, −6.57660785225189436758128727622, −6.07118664594875479479696825091, −4.80417154029467279067438775916, −4.10052303793859886411899334045, −3.21992246693943485427719128222, −1.52977845435573812011196719309, 0,
1.52977845435573812011196719309, 3.21992246693943485427719128222, 4.10052303793859886411899334045, 4.80417154029467279067438775916, 6.07118664594875479479696825091, 6.57660785225189436758128727622, 7.74963319311631313863943114023, 8.372034008912819522598019778511, 8.964905150568887964535101070416
