| L(s) = 1 | + 1.18·3-s + 5-s + 1.34·7-s − 1.59·9-s + 4.11·11-s + 4.12·13-s + 1.18·15-s − 0.563·17-s + 3.58·19-s + 1.59·21-s − 7.93·23-s + 25-s − 5.44·27-s − 0.950·29-s + 5.09·31-s + 4.87·33-s + 1.34·35-s − 0.121·37-s + 4.88·39-s + 7.89·41-s − 1.45·43-s − 1.59·45-s + 7.26·47-s − 5.19·49-s − 0.668·51-s + 7.53·53-s + 4.11·55-s + ⋯ |
| L(s) = 1 | + 0.684·3-s + 0.447·5-s + 0.508·7-s − 0.531·9-s + 1.24·11-s + 1.14·13-s + 0.306·15-s − 0.136·17-s + 0.823·19-s + 0.347·21-s − 1.65·23-s + 0.200·25-s − 1.04·27-s − 0.176·29-s + 0.915·31-s + 0.848·33-s + 0.227·35-s − 0.0200·37-s + 0.782·39-s + 1.23·41-s − 0.221·43-s − 0.237·45-s + 1.05·47-s − 0.741·49-s − 0.0935·51-s + 1.03·53-s + 0.554·55-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
| \(L(1)\) |
\(\approx\) |
\(3.382993106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.382993106\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.18T + 3T^{2} \) |
| 7 | \( 1 - 1.34T + 7T^{2} \) |
| 11 | \( 1 - 4.11T + 11T^{2} \) |
| 13 | \( 1 - 4.12T + 13T^{2} \) |
| 17 | \( 1 + 0.563T + 17T^{2} \) |
| 19 | \( 1 - 3.58T + 19T^{2} \) |
| 23 | \( 1 + 7.93T + 23T^{2} \) |
| 29 | \( 1 + 0.950T + 29T^{2} \) |
| 31 | \( 1 - 5.09T + 31T^{2} \) |
| 37 | \( 1 + 0.121T + 37T^{2} \) |
| 41 | \( 1 - 7.89T + 41T^{2} \) |
| 43 | \( 1 + 1.45T + 43T^{2} \) |
| 47 | \( 1 - 7.26T + 47T^{2} \) |
| 53 | \( 1 - 7.53T + 53T^{2} \) |
| 59 | \( 1 - 8.56T + 59T^{2} \) |
| 61 | \( 1 + 0.171T + 61T^{2} \) |
| 67 | \( 1 + 0.197T + 67T^{2} \) |
| 71 | \( 1 - 1.03T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 4.90T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 - 0.435T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−8.255445856773715520217357450958, −7.51775661301643300684170177435, −6.55410884556462155874507928235, −5.98333633508880198690336706929, −5.34873648841236533830213134564, −4.14377726812854216231210103299, −3.74707989940923522829027842693, −2.72793215269975870299169748005, −1.88644118752605178831599244770, −0.994813597413007908953938463820,
0.994813597413007908953938463820, 1.88644118752605178831599244770, 2.72793215269975870299169748005, 3.74707989940923522829027842693, 4.14377726812854216231210103299, 5.34873648841236533830213134564, 5.98333633508880198690336706929, 6.55410884556462155874507928235, 7.51775661301643300684170177435, 8.255445856773715520217357450958
