| L(s) = 1 | + 2.06·3-s − 4.30·5-s + 1.72·7-s + 1.24·9-s + 4.37·11-s − 4.54·13-s − 8.88·15-s − 3.59·19-s + 3.56·21-s − 3.79·23-s + 13.5·25-s − 3.61·27-s + 0.524·29-s − 8.07·31-s + 9.01·33-s − 7.44·35-s − 2.24·37-s − 9.35·39-s − 7.65·41-s + 2.25·43-s − 5.37·45-s − 1.33·47-s − 4.01·49-s + 12.5·53-s − 18.8·55-s − 7.41·57-s + 5.17·59-s + ⋯ |
| L(s) = 1 | + 1.18·3-s − 1.92·5-s + 0.653·7-s + 0.415·9-s + 1.31·11-s − 1.25·13-s − 2.29·15-s − 0.825·19-s + 0.777·21-s − 0.790·23-s + 2.71·25-s − 0.694·27-s + 0.0973·29-s − 1.45·31-s + 1.56·33-s − 1.25·35-s − 0.368·37-s − 1.49·39-s − 1.19·41-s + 0.343·43-s − 0.801·45-s − 0.194·47-s − 0.573·49-s + 1.72·53-s − 2.54·55-s − 0.982·57-s + 0.673·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2.06T + 3T^{2} \) |
| 5 | \( 1 + 4.30T + 5T^{2} \) |
| 7 | \( 1 - 1.72T + 7T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 13 | \( 1 + 4.54T + 13T^{2} \) |
| 19 | \( 1 + 3.59T + 19T^{2} \) |
| 23 | \( 1 + 3.79T + 23T^{2} \) |
| 29 | \( 1 - 0.524T + 29T^{2} \) |
| 31 | \( 1 + 8.07T + 31T^{2} \) |
| 37 | \( 1 + 2.24T + 37T^{2} \) |
| 41 | \( 1 + 7.65T + 41T^{2} \) |
| 43 | \( 1 - 2.25T + 43T^{2} \) |
| 47 | \( 1 + 1.33T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 + 4.75T + 61T^{2} \) |
| 67 | \( 1 + 5.10T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 1.27T + 73T^{2} \) |
| 79 | \( 1 - 0.254T + 79T^{2} \) |
| 83 | \( 1 - 5.55T + 83T^{2} \) |
| 89 | \( 1 + 0.319T + 89T^{2} \) |
| 97 | \( 1 + 6.35T + 97T^{2} \) |
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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
−8.689211944887846131219500952397, −7.82498674564057730880113886613, −7.42454299887001941975714601317, −6.65117832654923771195443155062, −5.16708306871987119543500293178, −4.14900495616819107987452149138, −3.86431537755821520782565674489, −2.86933436892113972060670066697, −1.73538439103182756789332786346, 0,
1.73538439103182756789332786346, 2.86933436892113972060670066697, 3.86431537755821520782565674489, 4.14900495616819107987452149138, 5.16708306871987119543500293178, 6.65117832654923771195443155062, 7.42454299887001941975714601317, 7.82498674564057730880113886613, 8.689211944887846131219500952397
