| L(s) = 1 | + 2.34·2-s + 3.48·4-s + 3.58·5-s − 1.15·7-s + 3.46·8-s + 8.39·10-s + 6.34·13-s − 2.69·14-s + 1.15·16-s + 3.91·17-s − 1.48·19-s + 12.4·20-s − 1.24·23-s + 7.87·25-s + 14.8·26-s − 4.00·28-s − 5.63·29-s − 3.21·31-s − 4.23·32-s + 9.15·34-s − 4.12·35-s − 10.0·37-s − 3.46·38-s + 12.4·40-s + 1.94·41-s + 9.18·43-s − 2.91·46-s + ⋯ |
| L(s) = 1 | + 1.65·2-s + 1.74·4-s + 1.60·5-s − 0.434·7-s + 1.22·8-s + 2.65·10-s + 1.75·13-s − 0.720·14-s + 0.287·16-s + 0.948·17-s − 0.339·19-s + 2.79·20-s − 0.260·23-s + 1.57·25-s + 2.91·26-s − 0.756·28-s − 1.04·29-s − 0.577·31-s − 0.748·32-s + 1.57·34-s − 0.697·35-s − 1.64·37-s − 0.562·38-s + 1.96·40-s + 0.303·41-s + 1.40·43-s − 0.430·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.817674041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.817674041\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.34T + 2T^{2} \) |
| 5 | \( 1 - 3.58T + 5T^{2} \) |
| 7 | \( 1 + 1.15T + 7T^{2} \) |
| 13 | \( 1 - 6.34T + 13T^{2} \) |
| 17 | \( 1 - 3.91T + 17T^{2} \) |
| 19 | \( 1 + 1.48T + 19T^{2} \) |
| 23 | \( 1 + 1.24T + 23T^{2} \) |
| 29 | \( 1 + 5.63T + 29T^{2} \) |
| 31 | \( 1 + 3.21T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 - 1.94T + 41T^{2} \) |
| 43 | \( 1 - 9.18T + 43T^{2} \) |
| 47 | \( 1 - 0.248T + 47T^{2} \) |
| 53 | \( 1 - 5.82T + 53T^{2} \) |
| 59 | \( 1 + 6.60T + 59T^{2} \) |
| 61 | \( 1 - 3.29T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 - 5.26T + 71T^{2} \) |
| 73 | \( 1 + 2.21T + 73T^{2} \) |
| 79 | \( 1 - 2.96T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 8.93T + 89T^{2} \) |
| 97 | \( 1 - 9.88T + 97T^{2} \) |
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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
−8.857892420376957485935690199267, −7.60155646180908259484120400671, −6.64155392707198110047716500262, −6.02442445752622506474722037885, −5.73685476853592000214417527824, −4.98721798153592978924122676583, −3.82674818316626073885226712771, −3.32375106882978867411552644950, −2.26251258054052145681004312403, −1.45027794477842868383359188777,
1.45027794477842868383359188777, 2.26251258054052145681004312403, 3.32375106882978867411552644950, 3.82674818316626073885226712771, 4.98721798153592978924122676583, 5.73685476853592000214417527824, 6.02442445752622506474722037885, 6.64155392707198110047716500262, 7.60155646180908259484120400671, 8.857892420376957485935690199267
