L(s) = 1 | + 3.14·3-s + 2.58·5-s + 1.35·7-s + 6.86·9-s − 5.76·11-s − 3.53·13-s + 8.10·15-s + 7.42·17-s + 1.86·19-s + 4.25·21-s + 2.72·23-s + 1.65·25-s + 12.1·27-s + 4.67·29-s + 4.18·31-s − 18.0·33-s + 3.49·35-s − 3.82·37-s − 11.1·39-s − 4.08·41-s + 4.41·43-s + 17.7·45-s + 4.01·47-s − 5.16·49-s + 23.3·51-s − 12.7·53-s − 14.8·55-s + ⋯ |
L(s) = 1 | + 1.81·3-s + 1.15·5-s + 0.511·7-s + 2.28·9-s − 1.73·11-s − 0.980·13-s + 2.09·15-s + 1.80·17-s + 0.427·19-s + 0.927·21-s + 0.569·23-s + 0.331·25-s + 2.33·27-s + 0.867·29-s + 0.751·31-s − 3.14·33-s + 0.590·35-s − 0.629·37-s − 1.77·39-s − 0.638·41-s + 0.673·43-s + 2.63·45-s + 0.586·47-s − 0.738·49-s + 3.26·51-s − 1.75·53-s − 2.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.885135408\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.885135408\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 - 3.14T + 3T^{2} \) |
| 5 | \( 1 - 2.58T + 5T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 + 5.76T + 11T^{2} \) |
| 13 | \( 1 + 3.53T + 13T^{2} \) |
| 17 | \( 1 - 7.42T + 17T^{2} \) |
| 19 | \( 1 - 1.86T + 19T^{2} \) |
| 23 | \( 1 - 2.72T + 23T^{2} \) |
| 29 | \( 1 - 4.67T + 29T^{2} \) |
| 31 | \( 1 - 4.18T + 31T^{2} \) |
| 37 | \( 1 + 3.82T + 37T^{2} \) |
| 41 | \( 1 + 4.08T + 41T^{2} \) |
| 43 | \( 1 - 4.41T + 43T^{2} \) |
| 47 | \( 1 - 4.01T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 15.0T + 59T^{2} \) |
| 61 | \( 1 + 1.07T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 9.86T + 71T^{2} \) |
| 73 | \( 1 - 2.87T + 73T^{2} \) |
| 79 | \( 1 + 3.14T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 - 8.26T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−8.261636943497874579204326352694, −7.83871014334861143220175531356, −7.41346884684148103231290304347, −6.29900857214103128223032661971, −5.10201023395674580924681732430, −4.93885763931394005116089828723, −3.44890553265972911640941980097, −2.79841564344505548189509644010, −2.24500961416872146301892637494, −1.30057609268886392528917610996,
1.30057609268886392528917610996, 2.24500961416872146301892637494, 2.79841564344505548189509644010, 3.44890553265972911640941980097, 4.93885763931394005116089828723, 5.10201023395674580924681732430, 6.29900857214103128223032661971, 7.41346884684148103231290304347, 7.83871014334861143220175531356, 8.261636943497874579204326352694