| L(s) = 1 | − 0.349·2-s − 3-s − 1.87·4-s − 5-s + 0.349·6-s + 0.242·7-s + 1.35·8-s + 9-s + 0.349·10-s − 4.22·11-s + 1.87·12-s + 4.70·13-s − 0.0845·14-s + 15-s + 3.28·16-s − 2.28·17-s − 0.349·18-s + 4.01·19-s + 1.87·20-s − 0.242·21-s + 1.47·22-s + 5.05·23-s − 1.35·24-s + 25-s − 1.64·26-s − 27-s − 0.455·28-s + ⋯ |
| L(s) = 1 | − 0.246·2-s − 0.577·3-s − 0.939·4-s − 0.447·5-s + 0.142·6-s + 0.0915·7-s + 0.478·8-s + 0.333·9-s + 0.110·10-s − 1.27·11-s + 0.542·12-s + 1.30·13-s − 0.0226·14-s + 0.258·15-s + 0.820·16-s − 0.553·17-s − 0.0822·18-s + 0.920·19-s + 0.419·20-s − 0.0528·21-s + 0.314·22-s + 1.05·23-s − 0.276·24-s + 0.200·25-s − 0.321·26-s − 0.192·27-s − 0.0860·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7797595371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7797595371\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 0.349T + 2T^{2} \) |
| 7 | \( 1 - 0.242T + 7T^{2} \) |
| 11 | \( 1 + 4.22T + 11T^{2} \) |
| 13 | \( 1 - 4.70T + 13T^{2} \) |
| 17 | \( 1 + 2.28T + 17T^{2} \) |
| 19 | \( 1 - 4.01T + 19T^{2} \) |
| 23 | \( 1 - 5.05T + 23T^{2} \) |
| 29 | \( 1 - 2.72T + 29T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 + 4.17T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 8.27T + 43T^{2} \) |
| 47 | \( 1 - 4.35T + 47T^{2} \) |
| 53 | \( 1 + 2.20T + 53T^{2} \) |
| 59 | \( 1 + 3.80T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 0.929T + 67T^{2} \) |
| 71 | \( 1 - 6.02T + 71T^{2} \) |
| 73 | \( 1 + 5.13T + 73T^{2} \) |
| 79 | \( 1 - 4.07T + 79T^{2} \) |
| 83 | \( 1 + 1.18T + 83T^{2} \) |
| 89 | \( 1 - 18.2T + 89T^{2} \) |
| 97 | \( 1 - 2.65T + 97T^{2} \) |
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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
−7.984098593986270284575567498371, −7.61357748210230505927449860818, −6.67452391962387628901824962943, −5.81093850015588109337528535722, −5.11330549034609192890383206644, −4.62728105967687517023951616415, −3.70231672398150822367812128443, −2.96807720333178272715250387065, −1.49331751040063762410611312443, −0.53460851923852027095125755250,
0.53460851923852027095125755250, 1.49331751040063762410611312443, 2.96807720333178272715250387065, 3.70231672398150822367812128443, 4.62728105967687517023951616415, 5.11330549034609192890383206644, 5.81093850015588109337528535722, 6.67452391962387628901824962943, 7.61357748210230505927449860818, 7.984098593986270284575567498371
