| L(s) = 1 | + 2.62·3-s + 1.15·5-s + 2.51·7-s + 3.88·9-s + 6.29·11-s + 0.699·13-s + 3.01·15-s + 4.58·17-s − 7.23·19-s + 6.59·21-s + 4.43·23-s − 3.67·25-s + 2.31·27-s + 3.16·29-s − 6.86·31-s + 16.5·33-s + 2.89·35-s + 3.34·37-s + 1.83·39-s + 1.17·41-s − 8.14·43-s + 4.46·45-s − 7.94·47-s − 0.678·49-s + 12.0·51-s + 6.14·53-s + 7.24·55-s + ⋯ |
| L(s) = 1 | + 1.51·3-s + 0.514·5-s + 0.950·7-s + 1.29·9-s + 1.89·11-s + 0.193·13-s + 0.779·15-s + 1.11·17-s − 1.65·19-s + 1.43·21-s + 0.925·23-s − 0.735·25-s + 0.446·27-s + 0.588·29-s − 1.23·31-s + 2.87·33-s + 0.488·35-s + 0.549·37-s + 0.293·39-s + 0.184·41-s − 1.24·43-s + 0.665·45-s − 1.15·47-s − 0.0969·49-s + 1.68·51-s + 0.844·53-s + 0.977·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.737429571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.737429571\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.62T + 3T^{2} \) |
| 5 | \( 1 - 1.15T + 5T^{2} \) |
| 7 | \( 1 - 2.51T + 7T^{2} \) |
| 11 | \( 1 - 6.29T + 11T^{2} \) |
| 13 | \( 1 - 0.699T + 13T^{2} \) |
| 17 | \( 1 - 4.58T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 - 4.43T + 23T^{2} \) |
| 29 | \( 1 - 3.16T + 29T^{2} \) |
| 31 | \( 1 + 6.86T + 31T^{2} \) |
| 37 | \( 1 - 3.34T + 37T^{2} \) |
| 41 | \( 1 - 1.17T + 41T^{2} \) |
| 43 | \( 1 + 8.14T + 43T^{2} \) |
| 47 | \( 1 + 7.94T + 47T^{2} \) |
| 53 | \( 1 - 6.14T + 53T^{2} \) |
| 59 | \( 1 + 3.29T + 59T^{2} \) |
| 61 | \( 1 - 1.51T + 61T^{2} \) |
| 67 | \( 1 + 4.60T + 67T^{2} \) |
| 71 | \( 1 - 8.22T + 71T^{2} \) |
| 73 | \( 1 + 5.16T + 73T^{2} \) |
| 79 | \( 1 + 8.65T + 79T^{2} \) |
| 83 | \( 1 + 4.65T + 83T^{2} \) |
| 89 | \( 1 - 3.59T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−8.434077045525154533269002915656, −8.011860729072387025534123044475, −7.05645717178985819759950254345, −6.40923072309108674153685293582, −5.45080451998161556641359653685, −4.35934953779634036569075458040, −3.82288211836202672298358916633, −2.95203979214546106969841317900, −1.83823241666389642066159247518, −1.42923146110699954661768062415,
1.42923146110699954661768062415, 1.83823241666389642066159247518, 2.95203979214546106969841317900, 3.82288211836202672298358916633, 4.35934953779634036569075458040, 5.45080451998161556641359653685, 6.40923072309108674153685293582, 7.05645717178985819759950254345, 8.011860729072387025534123044475, 8.434077045525154533269002915656
