| L(s) = 1 | − 0.653·3-s + 5-s − 3.25·7-s − 2.57·9-s + 0.974·11-s + 0.528·13-s − 0.653·15-s − 2.06·17-s − 0.748·19-s + 2.12·21-s − 23-s + 25-s + 3.64·27-s − 2.72·29-s + 9.42·31-s − 0.637·33-s − 3.25·35-s + 10.1·37-s − 0.345·39-s + 10.6·41-s + 2.90·43-s − 2.57·45-s − 0.472·47-s + 3.57·49-s + 1.35·51-s + 8.08·53-s + 0.974·55-s + ⋯ |
| L(s) = 1 | − 0.377·3-s + 0.447·5-s − 1.22·7-s − 0.857·9-s + 0.293·11-s + 0.146·13-s − 0.168·15-s − 0.501·17-s − 0.171·19-s + 0.463·21-s − 0.208·23-s + 0.200·25-s + 0.700·27-s − 0.505·29-s + 1.69·31-s − 0.110·33-s − 0.549·35-s + 1.66·37-s − 0.0552·39-s + 1.65·41-s + 0.443·43-s − 0.383·45-s − 0.0689·47-s + 0.510·49-s + 0.189·51-s + 1.11·53-s + 0.131·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 0.653T + 3T^{2} \) |
| 7 | \( 1 + 3.25T + 7T^{2} \) |
| 11 | \( 1 - 0.974T + 11T^{2} \) |
| 13 | \( 1 - 0.528T + 13T^{2} \) |
| 17 | \( 1 + 2.06T + 17T^{2} \) |
| 19 | \( 1 + 0.748T + 19T^{2} \) |
| 29 | \( 1 + 2.72T + 29T^{2} \) |
| 31 | \( 1 - 9.42T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 2.90T + 43T^{2} \) |
| 47 | \( 1 + 0.472T + 47T^{2} \) |
| 53 | \( 1 - 8.08T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 3.08T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 6.39T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 8.99T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 3.22T + 97T^{2} \) |
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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
−7.47541877011088005322352172618, −6.63535442351765160723828725095, −6.01159203186662067418956185641, −5.86212254024992124821331947017, −4.67444245570657512433393781070, −4.00907844812684680131573871541, −2.91407072990279183467499136134, −2.53379397899035781716252605014, −1.10545881510555337890882002271, 0,
1.10545881510555337890882002271, 2.53379397899035781716252605014, 2.91407072990279183467499136134, 4.00907844812684680131573871541, 4.67444245570657512433393781070, 5.86212254024992124821331947017, 6.01159203186662067418956185641, 6.63535442351765160723828725095, 7.47541877011088005322352172618
