| L(s) = 1 | + 1.58·2-s + 0.519·4-s − 5-s + 2.72·7-s − 2.35·8-s − 1.58·10-s + 0.148·11-s + 13-s + 4.32·14-s − 4.76·16-s + 5.13·17-s + 7.28·19-s − 0.519·20-s + 0.236·22-s − 5.32·23-s + 25-s + 1.58·26-s + 1.41·28-s − 4.08·29-s + 8.19·31-s − 2.86·32-s + 8.15·34-s − 2.72·35-s + 7.47·37-s + 11.5·38-s + 2.35·40-s + 8.59·41-s + ⋯ |
| L(s) = 1 | + 1.12·2-s + 0.259·4-s − 0.447·5-s + 1.02·7-s − 0.830·8-s − 0.501·10-s + 0.0448·11-s + 0.277·13-s + 1.15·14-s − 1.19·16-s + 1.24·17-s + 1.67·19-s − 0.116·20-s + 0.0503·22-s − 1.11·23-s + 0.200·25-s + 0.311·26-s + 0.267·28-s − 0.758·29-s + 1.47·31-s − 0.507·32-s + 1.39·34-s − 0.460·35-s + 1.22·37-s + 1.87·38-s + 0.371·40-s + 1.34·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.010817197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.010817197\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 1.58T + 2T^{2} \) |
| 7 | \( 1 - 2.72T + 7T^{2} \) |
| 11 | \( 1 - 0.148T + 11T^{2} \) |
| 17 | \( 1 - 5.13T + 17T^{2} \) |
| 19 | \( 1 - 7.28T + 19T^{2} \) |
| 23 | \( 1 + 5.32T + 23T^{2} \) |
| 29 | \( 1 + 4.08T + 29T^{2} \) |
| 31 | \( 1 - 8.19T + 31T^{2} \) |
| 37 | \( 1 - 7.47T + 37T^{2} \) |
| 41 | \( 1 - 8.59T + 41T^{2} \) |
| 43 | \( 1 + 9.95T + 43T^{2} \) |
| 47 | \( 1 - 0.909T + 47T^{2} \) |
| 53 | \( 1 - 9.31T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 8.52T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 + 0.424T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + 5.39T + 79T^{2} \) |
| 83 | \( 1 + 4.32T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−9.393968371315309133130857066929, −8.174348307652292023879680304868, −7.889501810677538812377561641483, −6.77646373690526899671390071207, −5.68042182335304972765758838743, −5.25131105041287303402395299514, −4.29016896115038826802090638465, −3.62749737317916604714674110223, −2.62749775432967688437901949916, −1.08725933018361281519570945779,
1.08725933018361281519570945779, 2.62749775432967688437901949916, 3.62749737317916604714674110223, 4.29016896115038826802090638465, 5.25131105041287303402395299514, 5.68042182335304972765758838743, 6.77646373690526899671390071207, 7.889501810677538812377561641483, 8.174348307652292023879680304868, 9.393968371315309133130857066929
