| L(s) = 1 | − 2.61·3-s + 3.85·5-s + 0.108·7-s + 3.85·9-s + 1.90·11-s + 2.14·13-s − 10.0·15-s − 3.71·17-s − 5.45·19-s − 0.285·21-s + 2.88·23-s + 9.85·25-s − 2.23·27-s − 5.96·29-s − 2.15·31-s − 4.97·33-s + 0.419·35-s + 5.99·37-s − 5.60·39-s + 5.71·41-s − 9.10·43-s + 14.8·45-s − 3.93·47-s − 6.98·49-s + 9.71·51-s − 4.29·53-s + 7.32·55-s + ⋯ |
| L(s) = 1 | − 1.51·3-s + 1.72·5-s + 0.0411·7-s + 1.28·9-s + 0.573·11-s + 0.593·13-s − 2.60·15-s − 0.900·17-s − 1.25·19-s − 0.0622·21-s + 0.601·23-s + 1.97·25-s − 0.430·27-s − 1.10·29-s − 0.386·31-s − 0.866·33-s + 0.0709·35-s + 0.985·37-s − 0.897·39-s + 0.893·41-s − 1.38·43-s + 2.21·45-s − 0.574·47-s − 0.998·49-s + 1.36·51-s − 0.590·53-s + 0.987·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 - 3.85T + 5T^{2} \) |
| 7 | \( 1 - 0.108T + 7T^{2} \) |
| 11 | \( 1 - 1.90T + 11T^{2} \) |
| 13 | \( 1 - 2.14T + 13T^{2} \) |
| 17 | \( 1 + 3.71T + 17T^{2} \) |
| 19 | \( 1 + 5.45T + 19T^{2} \) |
| 23 | \( 1 - 2.88T + 23T^{2} \) |
| 29 | \( 1 + 5.96T + 29T^{2} \) |
| 31 | \( 1 + 2.15T + 31T^{2} \) |
| 37 | \( 1 - 5.99T + 37T^{2} \) |
| 41 | \( 1 - 5.71T + 41T^{2} \) |
| 43 | \( 1 + 9.10T + 43T^{2} \) |
| 47 | \( 1 + 3.93T + 47T^{2} \) |
| 53 | \( 1 + 4.29T + 53T^{2} \) |
| 59 | \( 1 + 9.93T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 + 0.921T + 67T^{2} \) |
| 71 | \( 1 + 8.16T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 6.51T + 79T^{2} \) |
| 83 | \( 1 + 4.88T + 83T^{2} \) |
| 89 | \( 1 - 1.95T + 89T^{2} \) |
| 97 | \( 1 + 0.592T + 97T^{2} \) |
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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
−7.03644300444103752573260853136, −6.53323398084424447809992689660, −6.04580502890654220733714626454, −5.66216465071043866011165736810, −4.79838668004448630910842162991, −4.27757832602199625890872508621, −2.98108949758091424968436868118, −1.87242162700271589593293705719, −1.35376357038034929074249733712, 0,
1.35376357038034929074249733712, 1.87242162700271589593293705719, 2.98108949758091424968436868118, 4.27757832602199625890872508621, 4.79838668004448630910842162991, 5.66216465071043866011165736810, 6.04580502890654220733714626454, 6.53323398084424447809992689660, 7.03644300444103752573260853136
