L(s) = 1 | − 2.34·2-s + 3-s + 3.51·4-s − 2.34·6-s + 0.312·7-s − 3.57·8-s + 9-s − 5.22·11-s + 3.51·12-s − 5.05·13-s − 0.735·14-s + 1.35·16-s − 5.46·17-s − 2.34·18-s − 6.52·19-s + 0.312·21-s + 12.2·22-s + 3.29·23-s − 3.57·24-s + 11.8·26-s + 27-s + 1.10·28-s − 1.60·29-s − 7.51·31-s + 3.96·32-s − 5.22·33-s + 12.8·34-s + ⋯ |
L(s) = 1 | − 1.66·2-s + 0.577·3-s + 1.75·4-s − 0.959·6-s + 0.118·7-s − 1.26·8-s + 0.333·9-s − 1.57·11-s + 1.01·12-s − 1.40·13-s − 0.196·14-s + 0.337·16-s − 1.32·17-s − 0.553·18-s − 1.49·19-s + 0.0682·21-s + 2.61·22-s + 0.686·23-s − 0.728·24-s + 2.32·26-s + 0.192·27-s + 0.208·28-s − 0.297·29-s − 1.34·31-s + 0.701·32-s − 0.909·33-s + 2.20·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3041195424\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3041195424\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 107 | \( 1 - T \) |
good | 2 | \( 1 + 2.34T + 2T^{2} \) |
| 7 | \( 1 - 0.312T + 7T^{2} \) |
| 11 | \( 1 + 5.22T + 11T^{2} \) |
| 13 | \( 1 + 5.05T + 13T^{2} \) |
| 17 | \( 1 + 5.46T + 17T^{2} \) |
| 19 | \( 1 + 6.52T + 19T^{2} \) |
| 23 | \( 1 - 3.29T + 23T^{2} \) |
| 29 | \( 1 + 1.60T + 29T^{2} \) |
| 31 | \( 1 + 7.51T + 31T^{2} \) |
| 37 | \( 1 - 3.07T + 37T^{2} \) |
| 41 | \( 1 - 5.44T + 41T^{2} \) |
| 43 | \( 1 + 6.50T + 43T^{2} \) |
| 47 | \( 1 - 9.29T + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 5.67T + 61T^{2} \) |
| 67 | \( 1 + 2.29T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 2.39T + 73T^{2} \) |
| 79 | \( 1 + 3.32T + 79T^{2} \) |
| 83 | \( 1 - 2.13T + 83T^{2} \) |
| 89 | \( 1 + 0.353T + 89T^{2} \) |
| 97 | \( 1 + 8.99T + 97T^{2} \) |
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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
−8.015258087483051749275709557851, −7.34331718139592666716122831789, −6.95325329382802415969603258986, −6.02811502862695485233477869677, −4.94484968868320975055679858320, −4.38643643546257669727637931500, −3.00335354913444101272838861111, −2.30220069146299223149368568253, −1.89351458002178022435012444190, −0.31569958908668529062729957522,
0.31569958908668529062729957522, 1.89351458002178022435012444190, 2.30220069146299223149368568253, 3.00335354913444101272838861111, 4.38643643546257669727637931500, 4.94484968868320975055679858320, 6.02811502862695485233477869677, 6.95325329382802415969603258986, 7.34331718139592666716122831789, 8.015258087483051749275709557851