| L(s) = 1 | + 1.41·3-s + 0.329·5-s − 2.82·7-s − 1.01·9-s + 0.647·11-s − 5.15·13-s + 0.464·15-s + 0.793·17-s − 4.24·19-s − 3.98·21-s − 1.35·23-s − 4.89·25-s − 5.65·27-s + 7.17·29-s − 0.216·31-s + 0.913·33-s − 0.929·35-s + 1.38·37-s − 7.26·39-s − 10.2·41-s − 9.36·43-s − 0.332·45-s + 8.80·47-s + 0.969·49-s + 1.11·51-s + 8.56·53-s + 0.213·55-s + ⋯ |
| L(s) = 1 | + 0.814·3-s + 0.147·5-s − 1.06·7-s − 0.336·9-s + 0.195·11-s − 1.42·13-s + 0.119·15-s + 0.192·17-s − 0.972·19-s − 0.868·21-s − 0.282·23-s − 0.978·25-s − 1.08·27-s + 1.33·29-s − 0.0388·31-s + 0.158·33-s − 0.157·35-s + 0.227·37-s − 1.16·39-s − 1.59·41-s − 1.42·43-s − 0.0495·45-s + 1.28·47-s + 0.138·49-s + 0.156·51-s + 1.17·53-s + 0.0287·55-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 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 - 0.329T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 0.647T + 11T^{2} \) |
| 13 | \( 1 + 5.15T + 13T^{2} \) |
| 17 | \( 1 - 0.793T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 + 1.35T + 23T^{2} \) |
| 29 | \( 1 - 7.17T + 29T^{2} \) |
| 31 | \( 1 + 0.216T + 31T^{2} \) |
| 37 | \( 1 - 1.38T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 9.36T + 43T^{2} \) |
| 47 | \( 1 - 8.80T + 47T^{2} \) |
| 53 | \( 1 - 8.56T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 9.46T + 61T^{2} \) |
| 67 | \( 1 + 5.91T + 67T^{2} \) |
| 71 | \( 1 + 2.07T + 71T^{2} \) |
| 73 | \( 1 + 9.19T + 73T^{2} \) |
| 79 | \( 1 - 2.65T + 79T^{2} \) |
| 83 | \( 1 - 9.39T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−9.211447314687817175374250133722, −8.495921946602266801934685046658, −7.67738833806747573296586224787, −6.76426335603301011139846993792, −6.02019595135882054245362018940, −4.92245957888732753277832544022, −3.80050060360430659359009983484, −2.91281321381495051086065125200, −2.10790310474945145263368411920, 0,
2.10790310474945145263368411920, 2.91281321381495051086065125200, 3.80050060360430659359009983484, 4.92245957888732753277832544022, 6.02019595135882054245362018940, 6.76426335603301011139846993792, 7.67738833806747573296586224787, 8.495921946602266801934685046658, 9.211447314687817175374250133722
