| L(s) = 1 | − 0.110·2-s − 1.98·4-s + 0.0761·5-s + 3.79·7-s + 0.441·8-s − 0.00841·10-s − 11-s + 2.86·13-s − 0.419·14-s + 3.92·16-s − 4.34·17-s + 7.98·19-s − 0.151·20-s + 0.110·22-s − 8.82·23-s − 4.99·25-s − 0.316·26-s − 7.54·28-s − 8.77·29-s + 2.15·31-s − 1.31·32-s + 0.480·34-s + 0.288·35-s − 9.28·37-s − 0.882·38-s + 0.0335·40-s + 3.66·41-s + ⋯ |
| L(s) = 1 | − 0.0782·2-s − 0.993·4-s + 0.0340·5-s + 1.43·7-s + 0.155·8-s − 0.00266·10-s − 0.301·11-s + 0.794·13-s − 0.112·14-s + 0.981·16-s − 1.05·17-s + 1.83·19-s − 0.0338·20-s + 0.0235·22-s − 1.84·23-s − 0.998·25-s − 0.0621·26-s − 1.42·28-s − 1.62·29-s + 0.386·31-s − 0.232·32-s + 0.0823·34-s + 0.0488·35-s − 1.52·37-s − 0.143·38-s + 0.00530·40-s + 0.572·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 0.110T + 2T^{2} \) |
| 5 | \( 1 - 0.0761T + 5T^{2} \) |
| 7 | \( 1 - 3.79T + 7T^{2} \) |
| 13 | \( 1 - 2.86T + 13T^{2} \) |
| 17 | \( 1 + 4.34T + 17T^{2} \) |
| 19 | \( 1 - 7.98T + 19T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 29 | \( 1 + 8.77T + 29T^{2} \) |
| 31 | \( 1 - 2.15T + 31T^{2} \) |
| 37 | \( 1 + 9.28T + 37T^{2} \) |
| 41 | \( 1 - 3.66T + 41T^{2} \) |
| 43 | \( 1 + 3.58T + 43T^{2} \) |
| 47 | \( 1 + 6.97T + 47T^{2} \) |
| 53 | \( 1 - 9.56T + 53T^{2} \) |
| 59 | \( 1 - 4.57T + 59T^{2} \) |
| 67 | \( 1 + 6.56T + 67T^{2} \) |
| 71 | \( 1 + 7.11T + 71T^{2} \) |
| 73 | \( 1 + 5.09T + 73T^{2} \) |
| 79 | \( 1 - 5.57T + 79T^{2} \) |
| 83 | \( 1 + 4.90T + 83T^{2} \) |
| 89 | \( 1 + 6.47T + 89T^{2} \) |
| 97 | \( 1 - 7.14T + 97T^{2} \) |
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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
−7.75838648524957438087721138618, −7.37351821640763993702828884868, −6.04401242198258910807049137194, −5.47929887454213521053815166878, −4.86201946887737760120481703243, −4.06764858865763222375185115687, −3.50122360812546746410477098440, −2.06550726374232698740974947733, −1.36359955978649678514011854071, 0,
1.36359955978649678514011854071, 2.06550726374232698740974947733, 3.50122360812546746410477098440, 4.06764858865763222375185115687, 4.86201946887737760120481703243, 5.47929887454213521053815166878, 6.04401242198258910807049137194, 7.37351821640763993702828884868, 7.75838648524957438087721138618
