| L(s) = 1 | − 2.86·3-s + 0.462·5-s + 7-s + 5.18·9-s − 2·11-s − 6.64·13-s − 1.32·15-s − 17-s + 4.64·19-s − 2.86·21-s + 5.72·23-s − 4.78·25-s − 6.24·27-s − 0.796·29-s − 0.184·31-s + 5.72·33-s + 0.462·35-s + 7.44·37-s + 19.0·39-s − 6.71·41-s + 5.50·43-s + 2.39·45-s + 5.72·47-s + 49-s + 2.86·51-s + 4.33·53-s − 0.925·55-s + ⋯ |
| L(s) = 1 | − 1.65·3-s + 0.206·5-s + 0.377·7-s + 1.72·9-s − 0.603·11-s − 1.84·13-s − 0.341·15-s − 0.242·17-s + 1.06·19-s − 0.624·21-s + 1.19·23-s − 0.957·25-s − 1.20·27-s − 0.147·29-s − 0.0330·31-s + 0.996·33-s + 0.0781·35-s + 1.22·37-s + 3.04·39-s − 1.04·41-s + 0.839·43-s + 0.357·45-s + 0.834·47-s + 0.142·49-s + 0.400·51-s + 0.595·53-s − 0.124·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2.86T + 3T^{2} \) |
| 5 | \( 1 - 0.462T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 6.64T + 13T^{2} \) |
| 19 | \( 1 - 4.64T + 19T^{2} \) |
| 23 | \( 1 - 5.72T + 23T^{2} \) |
| 29 | \( 1 + 0.796T + 29T^{2} \) |
| 31 | \( 1 + 0.184T + 31T^{2} \) |
| 37 | \( 1 - 7.44T + 37T^{2} \) |
| 41 | \( 1 + 6.71T + 41T^{2} \) |
| 43 | \( 1 - 5.50T + 43T^{2} \) |
| 47 | \( 1 - 5.72T + 47T^{2} \) |
| 53 | \( 1 - 4.33T + 53T^{2} \) |
| 59 | \( 1 - 0.646T + 59T^{2} \) |
| 61 | \( 1 + 4.86T + 61T^{2} \) |
| 67 | \( 1 + 15.7T + 67T^{2} \) |
| 71 | \( 1 - 0.556T + 71T^{2} \) |
| 73 | \( 1 + 4.73T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 6.36T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.34232638594565110909754729723, −6.91237958750604997656684993434, −5.86431424178231375638596966912, −5.50304933992580213337637674143, −4.81435933822960201284560913261, −4.40038609909755662978463709342, −3.03113622062427515629637918844, −2.13076633927988678615073528120, −0.990851500381288648914857692131, 0,
0.990851500381288648914857692131, 2.13076633927988678615073528120, 3.03113622062427515629637918844, 4.40038609909755662978463709342, 4.81435933822960201284560913261, 5.50304933992580213337637674143, 5.86431424178231375638596966912, 6.91237958750604997656684993434, 7.34232638594565110909754729723
