| L(s) = 1 | + 3.42·3-s + 3.59·5-s − 7-s + 8.70·9-s − 1.08·11-s + 1.79·13-s + 12.2·15-s − 5.65·17-s + 0.0630·19-s − 3.42·21-s − 1.46·23-s + 7.91·25-s + 19.5·27-s + 5.04·29-s + 4.75·31-s − 3.70·33-s − 3.59·35-s + 7.19·37-s + 6.12·39-s − 7.50·41-s − 4.56·43-s + 31.2·45-s + 1.52·47-s + 49-s − 19.3·51-s + 6.59·53-s − 3.88·55-s + ⋯ |
| L(s) = 1 | + 1.97·3-s + 1.60·5-s − 0.377·7-s + 2.90·9-s − 0.326·11-s + 0.496·13-s + 3.17·15-s − 1.37·17-s + 0.0144·19-s − 0.746·21-s − 0.305·23-s + 1.58·25-s + 3.75·27-s + 0.936·29-s + 0.853·31-s − 0.644·33-s − 0.607·35-s + 1.18·37-s + 0.980·39-s − 1.17·41-s − 0.695·43-s + 4.66·45-s + 0.222·47-s + 0.142·49-s − 2.70·51-s + 0.905·53-s − 0.524·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.156489560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.156489560\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 3.42T + 3T^{2} \) |
| 5 | \( 1 - 3.59T + 5T^{2} \) |
| 11 | \( 1 + 1.08T + 11T^{2} \) |
| 13 | \( 1 - 1.79T + 13T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 - 0.0630T + 19T^{2} \) |
| 23 | \( 1 + 1.46T + 23T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 - 4.75T + 31T^{2} \) |
| 37 | \( 1 - 7.19T + 37T^{2} \) |
| 41 | \( 1 + 7.50T + 41T^{2} \) |
| 43 | \( 1 + 4.56T + 43T^{2} \) |
| 47 | \( 1 - 1.52T + 47T^{2} \) |
| 53 | \( 1 - 6.59T + 53T^{2} \) |
| 59 | \( 1 + 7.62T + 59T^{2} \) |
| 61 | \( 1 + 9.62T + 61T^{2} \) |
| 67 | \( 1 + 6.97T + 67T^{2} \) |
| 71 | \( 1 - 6.19T + 71T^{2} \) |
| 73 | \( 1 - 8.59T + 73T^{2} \) |
| 79 | \( 1 + 7.84T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 9.32T + 89T^{2} \) |
| 97 | \( 1 - 0.485T + 97T^{2} \) |
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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
−8.248441206276313271826361879448, −7.25331824919258420512989355565, −6.57858432306832806851762986742, −6.08458308497856276883831866503, −4.87994992841711582950643523445, −4.26974351521173113089300400785, −3.24887711610657866074317217431, −2.62126998771708787225951953892, −2.07813094145600871741811587390, −1.27068693734392346129915572566,
1.27068693734392346129915572566, 2.07813094145600871741811587390, 2.62126998771708787225951953892, 3.24887711610657866074317217431, 4.26974351521173113089300400785, 4.87994992841711582950643523445, 6.08458308497856276883831866503, 6.57858432306832806851762986742, 7.25331824919258420512989355565, 8.248441206276313271826361879448
