| L(s) = 1 | + 1.97·3-s + 5-s − 0.188·7-s + 0.892·9-s + 1.45·11-s − 5.70·13-s + 1.97·15-s − 3.29·17-s − 2.35·19-s − 0.371·21-s − 0.811·23-s + 25-s − 4.15·27-s + 6.35·29-s − 1.00·31-s + 2.87·33-s − 0.188·35-s − 6.49·37-s − 11.2·39-s − 9.03·41-s − 10.4·43-s + 0.892·45-s + 7.03·47-s − 6.96·49-s − 6.50·51-s + 5.21·53-s + 1.45·55-s + ⋯ |
| L(s) = 1 | + 1.13·3-s + 0.447·5-s − 0.0712·7-s + 0.297·9-s + 0.439·11-s − 1.58·13-s + 0.509·15-s − 0.799·17-s − 0.540·19-s − 0.0811·21-s − 0.169·23-s + 0.200·25-s − 0.800·27-s + 1.17·29-s − 0.180·31-s + 0.500·33-s − 0.0318·35-s − 1.06·37-s − 1.80·39-s − 1.41·41-s − 1.58·43-s + 0.133·45-s + 1.02·47-s − 0.994·49-s − 0.910·51-s + 0.716·53-s + 0.196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 - 1.97T + 3T^{2} \) |
| 7 | \( 1 + 0.188T + 7T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 13 | \( 1 + 5.70T + 13T^{2} \) |
| 17 | \( 1 + 3.29T + 17T^{2} \) |
| 19 | \( 1 + 2.35T + 19T^{2} \) |
| 23 | \( 1 + 0.811T + 23T^{2} \) |
| 29 | \( 1 - 6.35T + 29T^{2} \) |
| 31 | \( 1 + 1.00T + 31T^{2} \) |
| 37 | \( 1 + 6.49T + 37T^{2} \) |
| 41 | \( 1 + 9.03T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 7.03T + 47T^{2} \) |
| 53 | \( 1 - 5.21T + 53T^{2} \) |
| 59 | \( 1 - 7.83T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 0.795T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 9.53T + 73T^{2} \) |
| 79 | \( 1 + 17.1T + 79T^{2} \) |
| 83 | \( 1 - 6.51T + 83T^{2} \) |
| 89 | \( 1 + 7.05T + 89T^{2} \) |
| 97 | \( 1 - 5.93T + 97T^{2} \) |
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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
−7.893756437930285092045074963174, −6.89100654912665959790232653930, −6.64617814194543569828441668986, −5.45099224154532415426252474582, −4.79176067029622717623813689390, −3.94007451606938330453389271232, −3.05271459057623122443468133851, −2.37492913390105938249132819726, −1.70393370706114242807146674208, 0,
1.70393370706114242807146674208, 2.37492913390105938249132819726, 3.05271459057623122443468133851, 3.94007451606938330453389271232, 4.79176067029622717623813689390, 5.45099224154532415426252474582, 6.64617814194543569828441668986, 6.89100654912665959790232653930, 7.893756437930285092045074963174
