| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s + 9-s + 3.26·11-s − 12-s − 2.73·13-s + 2·14-s + 16-s + 4.93·17-s − 18-s + 4.93·19-s + 2·21-s − 3.26·22-s + 0.521·23-s + 24-s + 2.73·26-s − 27-s − 2·28-s + 0.521·29-s − 31-s − 32-s − 3.26·33-s − 4.93·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 0.333·9-s + 0.983·11-s − 0.288·12-s − 0.759·13-s + 0.534·14-s + 0.250·16-s + 1.19·17-s − 0.235·18-s + 1.13·19-s + 0.436·21-s − 0.695·22-s + 0.108·23-s + 0.204·24-s + 0.537·26-s − 0.192·27-s − 0.377·28-s + 0.0968·29-s − 0.179·31-s − 0.176·32-s − 0.567·33-s − 0.845·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.012772077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.012772077\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 3.26T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 - 4.93T + 17T^{2} \) |
| 19 | \( 1 - 4.93T + 19T^{2} \) |
| 23 | \( 1 - 0.521T + 23T^{2} \) |
| 29 | \( 1 - 0.521T + 29T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 8.82T + 43T^{2} \) |
| 47 | \( 1 + 4.93T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 9.34T + 59T^{2} \) |
| 61 | \( 1 + 9.75T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 5.56T + 71T^{2} \) |
| 73 | \( 1 - 3.04T + 73T^{2} \) |
| 79 | \( 1 + 6.41T + 79T^{2} \) |
| 83 | \( 1 - 1.36T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 7.26T + 97T^{2} \) |
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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
−8.209064497264286700334475362011, −7.59174604127695001783029055667, −6.85302392465612692636374816777, −6.29572385855561449179275244115, −5.53520038971741542371087004209, −4.70707677864288064361336871533, −3.58507823811645053884986134088, −2.91931212057375102680141612124, −1.60293390913198760054675371725, −0.66831876971305387295147282183,
0.66831876971305387295147282183, 1.60293390913198760054675371725, 2.91931212057375102680141612124, 3.58507823811645053884986134088, 4.70707677864288064361336871533, 5.53520038971741542371087004209, 6.29572385855561449179275244115, 6.85302392465612692636374816777, 7.59174604127695001783029055667, 8.209064497264286700334475362011
