| L(s) = 1 | + 1.06·2-s − 3-s − 0.856·4-s − 2.68·5-s − 1.06·6-s + 1.02·7-s − 3.05·8-s + 9-s − 2.87·10-s − 5.48·11-s + 0.856·12-s − 4.05·13-s + 1.09·14-s + 2.68·15-s − 1.55·16-s − 17-s + 1.06·18-s + 5.30·19-s + 2.30·20-s − 1.02·21-s − 5.86·22-s − 7.24·23-s + 3.05·24-s + 2.23·25-s − 4.33·26-s − 27-s − 0.880·28-s + ⋯ |
| L(s) = 1 | + 0.756·2-s − 0.577·3-s − 0.428·4-s − 1.20·5-s − 0.436·6-s + 0.388·7-s − 1.07·8-s + 0.333·9-s − 0.909·10-s − 1.65·11-s + 0.247·12-s − 1.12·13-s + 0.293·14-s + 0.694·15-s − 0.388·16-s − 0.242·17-s + 0.252·18-s + 1.21·19-s + 0.515·20-s − 0.224·21-s − 1.25·22-s − 1.51·23-s + 0.623·24-s + 0.447·25-s − 0.850·26-s − 0.192·27-s − 0.166·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3209465584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3209465584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 1.06T + 2T^{2} \) |
| 5 | \( 1 + 2.68T + 5T^{2} \) |
| 7 | \( 1 - 1.02T + 7T^{2} \) |
| 11 | \( 1 + 5.48T + 11T^{2} \) |
| 13 | \( 1 + 4.05T + 13T^{2} \) |
| 19 | \( 1 - 5.30T + 19T^{2} \) |
| 23 | \( 1 + 7.24T + 23T^{2} \) |
| 29 | \( 1 + 7.56T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 0.872T + 37T^{2} \) |
| 41 | \( 1 - 2.54T + 41T^{2} \) |
| 43 | \( 1 - 7.71T + 43T^{2} \) |
| 47 | \( 1 - 1.60T + 47T^{2} \) |
| 53 | \( 1 - 3.09T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 4.20T + 61T^{2} \) |
| 67 | \( 1 - 9.58T + 67T^{2} \) |
| 71 | \( 1 + 8.78T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 9.31T + 89T^{2} \) |
| 97 | \( 1 + 3.52T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.062728301862569992353126133356, −7.73157720778744043413497926456, −7.15558201159195541234779745142, −5.76621736952557263717849595628, −5.41407499411331421251449987962, −4.69357012046104157489950992190, −4.02847316411165036560045085788, −3.24707924008430902607106196612, −2.17575984926719871694473849638, −0.27967353701640543691449171198,
0.27967353701640543691449171198, 2.17575984926719871694473849638, 3.24707924008430902607106196612, 4.02847316411165036560045085788, 4.69357012046104157489950992190, 5.41407499411331421251449987962, 5.76621736952557263717849595628, 7.15558201159195541234779745142, 7.73157720778744043413497926456, 8.062728301862569992353126133356
