| L(s) = 1 | + 2.71·3-s − 2.20·5-s − 2.45·7-s + 4.38·9-s − 0.626·11-s + 2.36·13-s − 5.98·15-s − 1.70·17-s − 1.16·19-s − 6.67·21-s + 4.01·23-s − 0.145·25-s + 3.75·27-s − 5.57·29-s − 1.20·31-s − 1.70·33-s + 5.41·35-s − 10.8·37-s + 6.42·39-s + 3.14·41-s + 3.97·43-s − 9.65·45-s − 2.37·47-s − 0.956·49-s − 4.62·51-s − 13.2·53-s + 1.38·55-s + ⋯ |
| L(s) = 1 | + 1.56·3-s − 0.985·5-s − 0.929·7-s + 1.46·9-s − 0.188·11-s + 0.655·13-s − 1.54·15-s − 0.412·17-s − 0.267·19-s − 1.45·21-s + 0.838·23-s − 0.0291·25-s + 0.722·27-s − 1.03·29-s − 0.215·31-s − 0.296·33-s + 0.915·35-s − 1.77·37-s + 1.02·39-s + 0.490·41-s + 0.605·43-s − 1.43·45-s − 0.346·47-s − 0.136·49-s − 0.647·51-s − 1.81·53-s + 0.186·55-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.71T + 3T^{2} \) |
| 5 | \( 1 + 2.20T + 5T^{2} \) |
| 7 | \( 1 + 2.45T + 7T^{2} \) |
| 11 | \( 1 + 0.626T + 11T^{2} \) |
| 13 | \( 1 - 2.36T + 13T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 + 1.16T + 19T^{2} \) |
| 23 | \( 1 - 4.01T + 23T^{2} \) |
| 29 | \( 1 + 5.57T + 29T^{2} \) |
| 31 | \( 1 + 1.20T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 3.14T + 41T^{2} \) |
| 43 | \( 1 - 3.97T + 43T^{2} \) |
| 47 | \( 1 + 2.37T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 - 2.59T + 59T^{2} \) |
| 61 | \( 1 - 0.891T + 61T^{2} \) |
| 67 | \( 1 + 2.69T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 + 6.11T + 73T^{2} \) |
| 79 | \( 1 + 1.82T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 8.93T + 89T^{2} \) |
| 97 | \( 1 + 4.22T + 97T^{2} \) |
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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
−8.084336301341800064656662331848, −7.52960084430874398315054445935, −6.89042810048453745590727653641, −6.01409398111733801392323005329, −4.80824798625896079985445902183, −3.82616904268446376649525314274, −3.47730098477175564045506662200, −2.72828348985198770198857301466, −1.65349169009833434393899870346, 0,
1.65349169009833434393899870346, 2.72828348985198770198857301466, 3.47730098477175564045506662200, 3.82616904268446376649525314274, 4.80824798625896079985445902183, 6.01409398111733801392323005329, 6.89042810048453745590727653641, 7.52960084430874398315054445935, 8.084336301341800064656662331848
