| L(s) = 1 | + 2-s + 4-s + 3.34·5-s + 1.09·7-s + 8-s + 3.34·10-s − 0.788·11-s − 3.98·13-s + 1.09·14-s + 16-s + 3.16·17-s + 2.35·19-s + 3.34·20-s − 0.788·22-s + 2.43·23-s + 6.17·25-s − 3.98·26-s + 1.09·28-s + 5.83·29-s + 7.12·31-s + 32-s + 3.16·34-s + 3.67·35-s + 4.00·37-s + 2.35·38-s + 3.34·40-s − 9.66·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.49·5-s + 0.415·7-s + 0.353·8-s + 1.05·10-s − 0.237·11-s − 1.10·13-s + 0.293·14-s + 0.250·16-s + 0.767·17-s + 0.541·19-s + 0.747·20-s − 0.168·22-s + 0.507·23-s + 1.23·25-s − 0.781·26-s + 0.207·28-s + 1.08·29-s + 1.27·31-s + 0.176·32-s + 0.542·34-s + 0.621·35-s + 0.657·37-s + 0.382·38-s + 0.528·40-s − 1.51·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.992902852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.992902852\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 - 3.34T + 5T^{2} \) |
| 7 | \( 1 - 1.09T + 7T^{2} \) |
| 11 | \( 1 + 0.788T + 11T^{2} \) |
| 13 | \( 1 + 3.98T + 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 19 | \( 1 - 2.35T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 - 5.83T + 29T^{2} \) |
| 31 | \( 1 - 7.12T + 31T^{2} \) |
| 37 | \( 1 - 4.00T + 37T^{2} \) |
| 41 | \( 1 + 9.66T + 41T^{2} \) |
| 43 | \( 1 - 1.90T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 4.08T + 53T^{2} \) |
| 59 | \( 1 - 3.34T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 1.62T + 67T^{2} \) |
| 71 | \( 1 - 8.55T + 71T^{2} \) |
| 73 | \( 1 + 8.82T + 73T^{2} \) |
| 79 | \( 1 - 4.14T + 79T^{2} \) |
| 83 | \( 1 - 3.47T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−7.81846155079397962241890781699, −6.83972419306773027339885754431, −6.44018346495953760916413623489, −5.57629054959660521427523713600, −5.03768357266917592987194641561, −4.65428133743399946333135654424, −3.32863454392934417654132738086, −2.68350062936341529639193125229, −1.95366963943081658081755542159, −1.04843236294126568789944709147,
1.04843236294126568789944709147, 1.95366963943081658081755542159, 2.68350062936341529639193125229, 3.32863454392934417654132738086, 4.65428133743399946333135654424, 5.03768357266917592987194641561, 5.57629054959660521427523713600, 6.44018346495953760916413623489, 6.83972419306773027339885754431, 7.81846155079397962241890781699
