| L(s) = 1 | + 2.39·2-s + 3.74·4-s + 4.10·5-s − 3.62·7-s + 4.16·8-s + 9.83·10-s + 1.86·11-s + 3.35·13-s − 8.68·14-s + 2.50·16-s + 0.173·17-s − 1.09·19-s + 15.3·20-s + 4.47·22-s − 5.06·23-s + 11.8·25-s + 8.04·26-s − 13.5·28-s + 6.94·29-s + 6.11·31-s − 2.32·32-s + 0.414·34-s − 14.8·35-s + 3.44·37-s − 2.63·38-s + 17.1·40-s − 5.66·41-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 1.87·4-s + 1.83·5-s − 1.36·7-s + 1.47·8-s + 3.11·10-s + 0.562·11-s + 0.931·13-s − 2.32·14-s + 0.627·16-s + 0.0419·17-s − 0.251·19-s + 3.43·20-s + 0.953·22-s − 1.05·23-s + 2.37·25-s + 1.57·26-s − 2.56·28-s + 1.28·29-s + 1.09·31-s − 0.411·32-s + 0.0711·34-s − 2.51·35-s + 0.566·37-s − 0.426·38-s + 2.70·40-s − 0.884·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.105678086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.105678086\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - T \) |
| good | 2 | \( 1 - 2.39T + 2T^{2} \) |
| 5 | \( 1 - 4.10T + 5T^{2} \) |
| 7 | \( 1 + 3.62T + 7T^{2} \) |
| 11 | \( 1 - 1.86T + 11T^{2} \) |
| 13 | \( 1 - 3.35T + 13T^{2} \) |
| 17 | \( 1 - 0.173T + 17T^{2} \) |
| 19 | \( 1 + 1.09T + 19T^{2} \) |
| 23 | \( 1 + 5.06T + 23T^{2} \) |
| 29 | \( 1 - 6.94T + 29T^{2} \) |
| 31 | \( 1 - 6.11T + 31T^{2} \) |
| 37 | \( 1 - 3.44T + 37T^{2} \) |
| 41 | \( 1 + 5.66T + 41T^{2} \) |
| 43 | \( 1 + 5.01T + 43T^{2} \) |
| 47 | \( 1 + 5.07T + 47T^{2} \) |
| 53 | \( 1 + 4.55T + 53T^{2} \) |
| 59 | \( 1 + 4.23T + 59T^{2} \) |
| 61 | \( 1 - 8.14T + 61T^{2} \) |
| 67 | \( 1 + 8.96T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 6.34T + 73T^{2} \) |
| 79 | \( 1 - 1.38T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−9.301070516860819017751572961830, −8.351582698616978488346451259980, −6.64930754937497013484357756086, −6.50012349437037220900422513260, −5.99762584649034792876380958690, −5.19621966167123245314589059251, −4.22061846216126462265015916947, −3.25683397441981352331110562582, −2.59097907066936945090329717861, −1.51550702145875351847690420241,
1.51550702145875351847690420241, 2.59097907066936945090329717861, 3.25683397441981352331110562582, 4.22061846216126462265015916947, 5.19621966167123245314589059251, 5.99762584649034792876380958690, 6.50012349437037220900422513260, 6.64930754937497013484357756086, 8.351582698616978488346451259980, 9.301070516860819017751572961830
