| L(s) = 1 | − 1.33·3-s + 2.46·5-s + 0.387·7-s − 1.20·9-s − 2.24·11-s + 5.28·13-s − 3.30·15-s − 17-s + 3.73·19-s − 0.519·21-s + 6.56·23-s + 1.08·25-s + 5.63·27-s + 0.661·29-s + 9.41·31-s + 2.99·33-s + 0.956·35-s + 1.52·37-s − 7.07·39-s − 6.02·41-s + 0.957·43-s − 2.98·45-s + 1.62·47-s − 6.84·49-s + 1.33·51-s − 13.1·53-s − 5.52·55-s + ⋯ |
| L(s) = 1 | − 0.772·3-s + 1.10·5-s + 0.146·7-s − 0.402·9-s − 0.675·11-s + 1.46·13-s − 0.852·15-s − 0.242·17-s + 0.855·19-s − 0.113·21-s + 1.36·23-s + 0.216·25-s + 1.08·27-s + 0.122·29-s + 1.69·31-s + 0.521·33-s + 0.161·35-s + 0.250·37-s − 1.13·39-s − 0.941·41-s + 0.145·43-s − 0.444·45-s + 0.237·47-s − 0.978·49-s + 0.187·51-s − 1.80·53-s − 0.744·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.034211171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.034211171\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 1.33T + 3T^{2} \) |
| 5 | \( 1 - 2.46T + 5T^{2} \) |
| 7 | \( 1 - 0.387T + 7T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 - 5.28T + 13T^{2} \) |
| 19 | \( 1 - 3.73T + 19T^{2} \) |
| 23 | \( 1 - 6.56T + 23T^{2} \) |
| 29 | \( 1 - 0.661T + 29T^{2} \) |
| 31 | \( 1 - 9.41T + 31T^{2} \) |
| 37 | \( 1 - 1.52T + 37T^{2} \) |
| 41 | \( 1 + 6.02T + 41T^{2} \) |
| 43 | \( 1 - 0.957T + 43T^{2} \) |
| 47 | \( 1 - 1.62T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 + 5.59T + 67T^{2} \) |
| 71 | \( 1 - 6.64T + 71T^{2} \) |
| 73 | \( 1 - 7.86T + 73T^{2} \) |
| 79 | \( 1 - 5.27T + 79T^{2} \) |
| 83 | \( 1 + 0.361T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 - 6.29T + 97T^{2} \) |
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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
−7.980272981887420407964038308841, −6.80491063403357541046730070545, −6.38062492176955076718011432650, −5.73656090187391931848808891038, −5.20256032245616613885534024616, −4.59636794645981787342487631097, −3.32273355962585023947457688656, −2.72696427108687688456960316105, −1.59987630002825852943794402110, −0.77586828343050576597757217752,
0.77586828343050576597757217752, 1.59987630002825852943794402110, 2.72696427108687688456960316105, 3.32273355962585023947457688656, 4.59636794645981787342487631097, 5.20256032245616613885534024616, 5.73656090187391931848808891038, 6.38062492176955076718011432650, 6.80491063403357541046730070545, 7.980272981887420407964038308841
