| L(s) = 1 | − 2.21·3-s − 5-s − 2.83·7-s + 1.90·9-s + 1.37·11-s + 4.28·13-s + 2.21·15-s − 3.33·17-s − 8.21·19-s + 6.28·21-s + 0.622·23-s + 25-s + 2.42·27-s − 5.18·29-s − 6.64·31-s − 3.05·33-s + 2.83·35-s − 37-s − 9.47·39-s + 2.42·41-s + 1.18·43-s − 1.90·45-s − 2.54·47-s + 1.04·49-s + 7.37·51-s − 2.56·53-s − 1.37·55-s + ⋯ |
| L(s) = 1 | − 1.27·3-s − 0.447·5-s − 1.07·7-s + 0.634·9-s + 0.415·11-s + 1.18·13-s + 0.571·15-s − 0.808·17-s − 1.88·19-s + 1.37·21-s + 0.129·23-s + 0.200·25-s + 0.467·27-s − 0.962·29-s − 1.19·31-s − 0.531·33-s + 0.479·35-s − 0.164·37-s − 1.51·39-s + 0.379·41-s + 0.180·43-s − 0.283·45-s − 0.370·47-s + 0.149·49-s + 1.03·51-s − 0.351·53-s − 0.185·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4581483129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4581483129\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + 2.21T + 3T^{2} \) |
| 7 | \( 1 + 2.83T + 7T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 - 4.28T + 13T^{2} \) |
| 17 | \( 1 + 3.33T + 17T^{2} \) |
| 19 | \( 1 + 8.21T + 19T^{2} \) |
| 23 | \( 1 - 0.622T + 23T^{2} \) |
| 29 | \( 1 + 5.18T + 29T^{2} \) |
| 31 | \( 1 + 6.64T + 31T^{2} \) |
| 41 | \( 1 - 2.42T + 41T^{2} \) |
| 43 | \( 1 - 1.18T + 43T^{2} \) |
| 47 | \( 1 + 2.54T + 47T^{2} \) |
| 53 | \( 1 + 2.56T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 8.87T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 - 0.622T + 73T^{2} \) |
| 79 | \( 1 - 9.26T + 79T^{2} \) |
| 83 | \( 1 - 3.16T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−8.910408193250787152051298257977, −7.963512889931907116290808075687, −6.89061658908650570175233282545, −6.31685349016461968721908142493, −6.00984202108683974402637706625, −4.89008622406770759882020627041, −4.05314150410369005689801752713, −3.34282286645308477242254172913, −1.89788937056822264997835145487, −0.42750024585458048502523172950,
0.42750024585458048502523172950, 1.89788937056822264997835145487, 3.34282286645308477242254172913, 4.05314150410369005689801752713, 4.89008622406770759882020627041, 6.00984202108683974402637706625, 6.31685349016461968721908142493, 6.89061658908650570175233282545, 7.963512889931907116290808075687, 8.910408193250787152051298257977
