| L(s) = 1 | − 0.470·2-s + 3-s − 1.77·4-s + 3.24·5-s − 0.470·6-s − 7-s + 1.77·8-s + 9-s − 1.52·10-s − 1.77·12-s − 2.47·13-s + 0.470·14-s + 3.24·15-s + 2.71·16-s − 3.24·17-s − 0.470·18-s − 5.30·19-s − 5.77·20-s − 21-s − 8.86·23-s + 1.77·24-s + 5.55·25-s + 1.16·26-s + 27-s + 1.77·28-s − 2.47·29-s − 1.52·30-s + ⋯ |
| L(s) = 1 | − 0.332·2-s + 0.577·3-s − 0.889·4-s + 1.45·5-s − 0.192·6-s − 0.377·7-s + 0.628·8-s + 0.333·9-s − 0.483·10-s − 0.513·12-s − 0.685·13-s + 0.125·14-s + 0.838·15-s + 0.679·16-s − 0.788·17-s − 0.110·18-s − 1.21·19-s − 1.29·20-s − 0.218·21-s − 1.84·23-s + 0.363·24-s + 1.11·25-s + 0.228·26-s + 0.192·27-s + 0.336·28-s − 0.458·29-s − 0.279·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.470T + 2T^{2} \) |
| 5 | \( 1 - 3.24T + 5T^{2} \) |
| 13 | \( 1 + 2.47T + 13T^{2} \) |
| 17 | \( 1 + 3.24T + 17T^{2} \) |
| 19 | \( 1 + 5.30T + 19T^{2} \) |
| 23 | \( 1 + 8.86T + 23T^{2} \) |
| 29 | \( 1 + 2.47T + 29T^{2} \) |
| 31 | \( 1 + 6.49T + 31T^{2} \) |
| 37 | \( 1 - 1.77T + 37T^{2} \) |
| 41 | \( 1 + 1.28T + 41T^{2} \) |
| 43 | \( 1 + 4.89T + 43T^{2} \) |
| 47 | \( 1 - 3.96T + 47T^{2} \) |
| 53 | \( 1 + 9.77T + 53T^{2} \) |
| 59 | \( 1 - 2.41T + 59T^{2} \) |
| 61 | \( 1 - 4.74T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 + 3.19T + 71T^{2} \) |
| 73 | \( 1 - 8.98T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 7.86T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−8.679638286319169319754914679811, −8.016032008781402413410157594126, −7.00990565423151717182272202693, −6.16997550867019494882356806177, −5.43534203127225120403183062608, −4.49496694656401828694853456336, −3.70438088034905898208980878375, −2.35368983806938460490585653915, −1.77293359957941054589938912673, 0,
1.77293359957941054589938912673, 2.35368983806938460490585653915, 3.70438088034905898208980878375, 4.49496694656401828694853456336, 5.43534203127225120403183062608, 6.16997550867019494882356806177, 7.00990565423151717182272202693, 8.016032008781402413410157594126, 8.679638286319169319754914679811
