| L(s) = 1 | + 0.640·2-s − 1.58·4-s − 1.64·7-s − 2.29·8-s + 2.17·11-s + 0.709·13-s − 1.05·14-s + 1.70·16-s − 3.14·17-s − 1.10·19-s + 1.39·22-s + 4.43·23-s + 0.454·26-s + 2.60·28-s − 3.71·29-s − 31-s + 5.69·32-s − 2.01·34-s + 2.16·37-s − 0.706·38-s + 3.53·41-s + 3.01·43-s − 3.45·44-s + 2.84·46-s + 5.33·47-s − 4.30·49-s − 1.12·52-s + ⋯ |
| L(s) = 1 | + 0.452·2-s − 0.794·4-s − 0.620·7-s − 0.812·8-s + 0.654·11-s + 0.196·13-s − 0.280·14-s + 0.426·16-s − 0.763·17-s − 0.253·19-s + 0.296·22-s + 0.925·23-s + 0.0891·26-s + 0.492·28-s − 0.690·29-s − 0.179·31-s + 1.00·32-s − 0.345·34-s + 0.356·37-s − 0.114·38-s + 0.552·41-s + 0.460·43-s − 0.520·44-s + 0.419·46-s + 0.777·47-s − 0.615·49-s − 0.156·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 0.640T + 2T^{2} \) |
| 7 | \( 1 + 1.64T + 7T^{2} \) |
| 11 | \( 1 - 2.17T + 11T^{2} \) |
| 13 | \( 1 - 0.709T + 13T^{2} \) |
| 17 | \( 1 + 3.14T + 17T^{2} \) |
| 19 | \( 1 + 1.10T + 19T^{2} \) |
| 23 | \( 1 - 4.43T + 23T^{2} \) |
| 29 | \( 1 + 3.71T + 29T^{2} \) |
| 37 | \( 1 - 2.16T + 37T^{2} \) |
| 41 | \( 1 - 3.53T + 41T^{2} \) |
| 43 | \( 1 - 3.01T + 43T^{2} \) |
| 47 | \( 1 - 5.33T + 47T^{2} \) |
| 53 | \( 1 - 9.33T + 53T^{2} \) |
| 59 | \( 1 + 9.45T + 59T^{2} \) |
| 61 | \( 1 - 5.70T + 61T^{2} \) |
| 67 | \( 1 + 6.19T + 67T^{2} \) |
| 71 | \( 1 - 4.77T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 5.23T + 83T^{2} \) |
| 89 | \( 1 - 1.03T + 89T^{2} \) |
| 97 | \( 1 + 0.848T + 97T^{2} \) |
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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
−7.51101402919523186692370101689, −6.79599693098889510212228871513, −6.08338960129734847078017422648, −5.51962722326629034844064691799, −4.58228944559477879292818919737, −4.06138710742613811713694784236, −3.33246715583209988737472377668, −2.49396124459371380385008285798, −1.17208830024103118126311230064, 0,
1.17208830024103118126311230064, 2.49396124459371380385008285798, 3.33246715583209988737472377668, 4.06138710742613811713694784236, 4.58228944559477879292818919737, 5.51962722326629034844064691799, 6.08338960129734847078017422648, 6.79599693098889510212228871513, 7.51101402919523186692370101689
