| L(s) = 1 | + 1.70·2-s + 0.894·4-s + 4.04·5-s + 4.09·7-s − 1.88·8-s + 6.87·10-s + 11-s + 13-s + 6.97·14-s − 4.98·16-s − 3.40·17-s − 0.396·19-s + 3.61·20-s + 1.70·22-s − 1.73·23-s + 11.3·25-s + 1.70·26-s + 3.66·28-s − 7.08·29-s − 8.11·31-s − 4.72·32-s − 5.78·34-s + 16.5·35-s + 2.42·37-s − 0.674·38-s − 7.60·40-s − 1.88·41-s + ⋯ |
| L(s) = 1 | + 1.20·2-s + 0.447·4-s + 1.80·5-s + 1.54·7-s − 0.665·8-s + 2.17·10-s + 0.301·11-s + 0.277·13-s + 1.86·14-s − 1.24·16-s − 0.825·17-s − 0.0909·19-s + 0.808·20-s + 0.362·22-s − 0.360·23-s + 2.26·25-s + 0.333·26-s + 0.692·28-s − 1.31·29-s − 1.45·31-s − 0.835·32-s − 0.992·34-s + 2.80·35-s + 0.399·37-s − 0.109·38-s − 1.20·40-s − 0.294·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.328148972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.328148972\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 1.70T + 2T^{2} \) |
| 5 | \( 1 - 4.04T + 5T^{2} \) |
| 7 | \( 1 - 4.09T + 7T^{2} \) |
| 17 | \( 1 + 3.40T + 17T^{2} \) |
| 19 | \( 1 + 0.396T + 19T^{2} \) |
| 23 | \( 1 + 1.73T + 23T^{2} \) |
| 29 | \( 1 + 7.08T + 29T^{2} \) |
| 31 | \( 1 + 8.11T + 31T^{2} \) |
| 37 | \( 1 - 2.42T + 37T^{2} \) |
| 41 | \( 1 + 1.88T + 41T^{2} \) |
| 43 | \( 1 - 2.07T + 43T^{2} \) |
| 47 | \( 1 + 3.47T + 47T^{2} \) |
| 53 | \( 1 + 6.80T + 53T^{2} \) |
| 59 | \( 1 + 3.45T + 59T^{2} \) |
| 61 | \( 1 - 4.40T + 61T^{2} \) |
| 67 | \( 1 + 1.82T + 67T^{2} \) |
| 71 | \( 1 - 2.68T + 71T^{2} \) |
| 73 | \( 1 - 9.07T + 73T^{2} \) |
| 79 | \( 1 - 9.08T + 79T^{2} \) |
| 83 | \( 1 - 9.37T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 5.24T + 97T^{2} \) |
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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
−9.447568653759186084393225402715, −9.098179019584828733177292300130, −8.059034454273887792018817904023, −6.81061206941000363846265157439, −6.01949418834818178888845711432, −5.34560343446388418151771438311, −4.78038903186610063736099961894, −3.74832914763498429887804307994, −2.32174377776056455699417094509, −1.67335524406448166930321215632,
1.67335524406448166930321215632, 2.32174377776056455699417094509, 3.74832914763498429887804307994, 4.78038903186610063736099961894, 5.34560343446388418151771438311, 6.01949418834818178888845711432, 6.81061206941000363846265157439, 8.059034454273887792018817904023, 9.098179019584828733177292300130, 9.447568653759186084393225402715
