| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 2·10-s − 5·11-s − 6·13-s + 16-s − 4·17-s + 4·19-s − 2·20-s + 5·22-s + 4·23-s − 25-s + 6·26-s − 7·29-s − 3·31-s − 32-s + 4·34-s + 8·37-s − 4·38-s + 2·40-s − 6·41-s + 8·43-s − 5·44-s − 4·46-s + 6·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s − 1.50·11-s − 1.66·13-s + 1/4·16-s − 0.970·17-s + 0.917·19-s − 0.447·20-s + 1.06·22-s + 0.834·23-s − 1/5·25-s + 1.17·26-s − 1.29·29-s − 0.538·31-s − 0.176·32-s + 0.685·34-s + 1.31·37-s − 0.648·38-s + 0.316·40-s − 0.937·41-s + 1.21·43-s − 0.753·44-s − 0.589·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4949434029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4949434029\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−8.852664685192944845022374486582, −7.948678631823869577486989243928, −7.41391959595930556776941898851, −7.08016768845276887296355816735, −5.67437679291396188991469247065, −5.02954632779517689480749219066, −4.05691084797417069928036854936, −2.87793724984651026489098410664, −2.21200891620018685371903520748, −0.45812635244474710229041310149,
0.45812635244474710229041310149, 2.21200891620018685371903520748, 2.87793724984651026489098410664, 4.05691084797417069928036854936, 5.02954632779517689480749219066, 5.67437679291396188991469247065, 7.08016768845276887296355816735, 7.41391959595930556776941898851, 7.948678631823869577486989243928, 8.852664685192944845022374486582
