| L(s) = 1 | + 2.52·2-s − 3-s + 4.35·4-s + 5-s − 2.52·6-s + 7-s + 5.92·8-s + 9-s + 2.52·10-s − 1.70·11-s − 4.35·12-s + 4.54·13-s + 2.52·14-s − 15-s + 6.23·16-s + 0.467·17-s + 2.52·18-s + 6.08·19-s + 4.35·20-s − 21-s − 4.30·22-s + 0.689·23-s − 5.92·24-s + 25-s + 11.4·26-s − 27-s + 4.35·28-s + ⋯ |
| L(s) = 1 | + 1.78·2-s − 0.577·3-s + 2.17·4-s + 0.447·5-s − 1.02·6-s + 0.377·7-s + 2.09·8-s + 0.333·9-s + 0.796·10-s − 0.514·11-s − 1.25·12-s + 1.26·13-s + 0.673·14-s − 0.258·15-s + 1.55·16-s + 0.113·17-s + 0.594·18-s + 1.39·19-s + 0.973·20-s − 0.218·21-s − 0.917·22-s + 0.143·23-s − 1.20·24-s + 0.200·25-s + 2.24·26-s − 0.192·27-s + 0.822·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.297633396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.297633396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 2.52T + 2T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 - 4.54T + 13T^{2} \) |
| 17 | \( 1 - 0.467T + 17T^{2} \) |
| 19 | \( 1 - 6.08T + 19T^{2} \) |
| 23 | \( 1 - 0.689T + 23T^{2} \) |
| 29 | \( 1 - 6.79T + 29T^{2} \) |
| 31 | \( 1 + 7.10T + 31T^{2} \) |
| 37 | \( 1 - 2.56T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 8.31T + 43T^{2} \) |
| 53 | \( 1 + 4.22T + 53T^{2} \) |
| 59 | \( 1 - 7.98T + 59T^{2} \) |
| 61 | \( 1 + 0.334T + 61T^{2} \) |
| 67 | \( 1 - 3.02T + 67T^{2} \) |
| 71 | \( 1 - 0.837T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 6.63T + 89T^{2} \) |
| 97 | \( 1 - 5.36T + 97T^{2} \) |
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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
−7.988526525606525216551039707839, −7.15645405460601340976247529817, −6.48151721529188631122156911764, −5.89823188604096384521960397933, −5.11543603491937575127349292884, −4.92819330381756649039932475682, −3.72630800944685689541867546420, −3.23816303218656702794178272521, −2.12935089383091336726584187875, −1.19105449886563366326315131050,
1.19105449886563366326315131050, 2.12935089383091336726584187875, 3.23816303218656702794178272521, 3.72630800944685689541867546420, 4.92819330381756649039932475682, 5.11543603491937575127349292884, 5.89823188604096384521960397933, 6.48151721529188631122156911764, 7.15645405460601340976247529817, 7.988526525606525216551039707839
