| L(s) = 1 | + 1.41·3-s − 5-s + 2.82·7-s − 0.999·9-s + 0.828·11-s + 3.41·13-s − 1.41·15-s + 4.82·17-s + 19-s + 4.00·21-s − 4·23-s + 25-s − 5.65·27-s − 0.828·29-s + 1.17·33-s − 2.82·35-s + 10.2·37-s + 4.82·39-s − 0.828·41-s − 2.82·43-s + 0.999·45-s + 8.48·47-s + 1.00·49-s + 6.82·51-s + 13.0·53-s − 0.828·55-s + 1.41·57-s + ⋯ |
| L(s) = 1 | + 0.816·3-s − 0.447·5-s + 1.06·7-s − 0.333·9-s + 0.249·11-s + 0.946·13-s − 0.365·15-s + 1.17·17-s + 0.229·19-s + 0.872·21-s − 0.834·23-s + 0.200·25-s − 1.08·27-s − 0.153·29-s + 0.203·33-s − 0.478·35-s + 1.68·37-s + 0.773·39-s − 0.129·41-s − 0.431·43-s + 0.149·45-s + 1.23·47-s + 0.142·49-s + 0.956·51-s + 1.79·53-s − 0.111·55-s + 0.187·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.431175751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.431175751\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 0.828T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 - 9.41T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 9.17T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 3.17T + 89T^{2} \) |
| 97 | \( 1 + 2.24T + 97T^{2} \) |
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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
−9.279899583000994898532622480357, −8.497137364131764817345041504282, −8.002401702181419939965463193504, −7.41891593420163043447895604177, −6.11906296075235778613866779405, −5.35388322309231451874930470982, −4.18096256886026010519908576510, −3.49860584015460536087637147757, −2.39117902522840006617867281474, −1.16703527292918390729128663720,
1.16703527292918390729128663720, 2.39117902522840006617867281474, 3.49860584015460536087637147757, 4.18096256886026010519908576510, 5.35388322309231451874930470982, 6.11906296075235778613866779405, 7.41891593420163043447895604177, 8.002401702181419939965463193504, 8.497137364131764817345041504282, 9.279899583000994898532622480357
