L(s) = 1 | − 1.40·2-s − 0.0197·4-s + 4.17·5-s − 0.442·7-s + 2.84·8-s − 5.86·10-s + 6.07·11-s − 5.08·13-s + 0.622·14-s − 3.96·16-s + 0.258·17-s + 19-s − 0.0825·20-s − 8.55·22-s + 2.95·23-s + 12.3·25-s + 7.15·26-s + 0.00874·28-s − 6.10·29-s − 9.36·31-s − 0.111·32-s − 0.363·34-s − 1.84·35-s + 9.75·37-s − 1.40·38-s + 11.8·40-s + 6.18·41-s + ⋯ |
L(s) = 1 | − 0.995·2-s − 0.00989·4-s + 1.86·5-s − 0.167·7-s + 1.00·8-s − 1.85·10-s + 1.83·11-s − 1.41·13-s + 0.166·14-s − 0.990·16-s + 0.0626·17-s + 0.229·19-s − 0.0184·20-s − 1.82·22-s + 0.615·23-s + 2.47·25-s + 1.40·26-s + 0.00165·28-s − 1.13·29-s − 1.68·31-s − 0.0197·32-s − 0.0623·34-s − 0.311·35-s + 1.60·37-s − 0.228·38-s + 1.87·40-s + 0.965·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.739299343\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.739299343\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 + 1.40T + 2T^{2} \) |
| 5 | \( 1 - 4.17T + 5T^{2} \) |
| 7 | \( 1 + 0.442T + 7T^{2} \) |
| 11 | \( 1 - 6.07T + 11T^{2} \) |
| 13 | \( 1 + 5.08T + 13T^{2} \) |
| 17 | \( 1 - 0.258T + 17T^{2} \) |
| 23 | \( 1 - 2.95T + 23T^{2} \) |
| 29 | \( 1 + 6.10T + 29T^{2} \) |
| 31 | \( 1 + 9.36T + 31T^{2} \) |
| 37 | \( 1 - 9.75T + 37T^{2} \) |
| 41 | \( 1 - 6.18T + 41T^{2} \) |
| 43 | \( 1 - 1.35T + 43T^{2} \) |
| 53 | \( 1 + 7.64T + 53T^{2} \) |
| 59 | \( 1 - 7.29T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 7.82T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 9.60T + 79T^{2} \) |
| 83 | \( 1 - 2.77T + 83T^{2} \) |
| 89 | \( 1 - 3.06T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−7.78062697486069468969290621464, −7.22743302546893992215543445675, −6.50786762772506234075925023233, −5.90446169090486644176499059020, −5.09351110991653617947476384891, −4.44948542814826762246133268477, −3.37384796427191087969689795299, −2.20899573480076698733879019858, −1.69167884163176866316919860912, −0.802944528252326578111070921496,
0.802944528252326578111070921496, 1.69167884163176866316919860912, 2.20899573480076698733879019858, 3.37384796427191087969689795299, 4.44948542814826762246133268477, 5.09351110991653617947476384891, 5.90446169090486644176499059020, 6.50786762772506234075925023233, 7.22743302546893992215543445675, 7.78062697486069468969290621464