L(s) = 1 | + 3-s + 2.72·5-s + 3.31·7-s + 9-s + 0.656·11-s + 2.72·15-s − 1.16·17-s + 6.77·19-s + 3.31·21-s + 6.90·23-s + 2.41·25-s + 27-s − 5.82·29-s − 0.969·31-s + 0.656·33-s + 9.02·35-s − 9.93·37-s + 10.5·41-s − 5.07·43-s + 2.72·45-s − 8.24·47-s + 3.99·49-s − 1.16·51-s + 0.841·53-s + 1.78·55-s + 6.77·57-s + 0.128·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.21·5-s + 1.25·7-s + 0.333·9-s + 0.197·11-s + 0.702·15-s − 0.283·17-s + 1.55·19-s + 0.723·21-s + 1.43·23-s + 0.482·25-s + 0.192·27-s − 1.08·29-s − 0.174·31-s + 0.114·33-s + 1.52·35-s − 1.63·37-s + 1.65·41-s − 0.773·43-s + 0.405·45-s − 1.20·47-s + 0.570·49-s − 0.163·51-s + 0.115·53-s + 0.241·55-s + 0.896·57-s + 0.0166·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.854063766\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.854063766\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 2.72T + 5T^{2} \) |
| 7 | \( 1 - 3.31T + 7T^{2} \) |
| 11 | \( 1 - 0.656T + 11T^{2} \) |
| 17 | \( 1 + 1.16T + 17T^{2} \) |
| 19 | \( 1 - 6.77T + 19T^{2} \) |
| 23 | \( 1 - 6.90T + 23T^{2} \) |
| 29 | \( 1 + 5.82T + 29T^{2} \) |
| 31 | \( 1 + 0.969T + 31T^{2} \) |
| 37 | \( 1 + 9.93T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 5.07T + 43T^{2} \) |
| 47 | \( 1 + 8.24T + 47T^{2} \) |
| 53 | \( 1 - 0.841T + 53T^{2} \) |
| 59 | \( 1 - 0.128T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 - 5.32T + 71T^{2} \) |
| 73 | \( 1 + 3.75T + 73T^{2} \) |
| 79 | \( 1 - 2.17T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 4.09T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−8.625482649295338227739327969820, −7.61253307417027621497342350423, −7.18409268476166208588916602616, −6.17655896972092836337074616176, −5.25684264035894549344661493865, −4.95109796859476632747362621707, −3.75031715504071794551370802059, −2.82368236504562179420383336656, −1.86312191754174594161417444205, −1.26540526978308159887043557136,
1.26540526978308159887043557136, 1.86312191754174594161417444205, 2.82368236504562179420383336656, 3.75031715504071794551370802059, 4.95109796859476632747362621707, 5.25684264035894549344661493865, 6.17655896972092836337074616176, 7.18409268476166208588916602616, 7.61253307417027621497342350423, 8.625482649295338227739327969820