L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 4.63·7-s + 8-s + 9-s + 5.23·11-s − 12-s + 3.39·13-s − 4.63·14-s + 16-s + 2.43·17-s + 18-s − 7.06·19-s + 4.63·21-s + 5.23·22-s − 3.43·23-s − 24-s + 3.39·26-s − 27-s − 4.63·28-s + 0.794·29-s + 31-s + 32-s − 5.23·33-s + 2.43·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.75·7-s + 0.353·8-s + 0.333·9-s + 1.57·11-s − 0.288·12-s + 0.942·13-s − 1.23·14-s + 0.250·16-s + 0.589·17-s + 0.235·18-s − 1.62·19-s + 1.01·21-s + 1.11·22-s − 0.715·23-s − 0.204·24-s + 0.666·26-s − 0.192·27-s − 0.876·28-s + 0.147·29-s + 0.179·31-s + 0.176·32-s − 0.912·33-s + 0.416·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.150278367\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.150278367\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + 4.63T + 7T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 - 3.39T + 13T^{2} \) |
| 17 | \( 1 - 2.43T + 17T^{2} \) |
| 19 | \( 1 + 7.06T + 19T^{2} \) |
| 23 | \( 1 + 3.43T + 23T^{2} \) |
| 29 | \( 1 - 0.794T + 29T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 4.41T + 41T^{2} \) |
| 43 | \( 1 + 4.63T + 43T^{2} \) |
| 47 | \( 1 + 1.56T + 47T^{2} \) |
| 53 | \( 1 - 9.43T + 53T^{2} \) |
| 59 | \( 1 - 8.06T + 59T^{2} \) |
| 61 | \( 1 - 8.43T + 61T^{2} \) |
| 67 | \( 1 + 7.46T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 3.22T + 83T^{2} \) |
| 89 | \( 1 - 5.36T + 89T^{2} \) |
| 97 | \( 1 - 0.191T + 97T^{2} \) |
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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
−8.404504717880127472182545369908, −7.13192995533801377154815530068, −6.60524088412105552835267510769, −6.14298758036090310788197703953, −5.67922576266729025632670205137, −4.31973525567357906125200260301, −3.88141202617894263215127672195, −3.20155603511879375977505706504, −1.99236157014802370329029037293, −0.75301695746337248569684513679,
0.75301695746337248569684513679, 1.99236157014802370329029037293, 3.20155603511879375977505706504, 3.88141202617894263215127672195, 4.31973525567357906125200260301, 5.67922576266729025632670205137, 6.14298758036090310788197703953, 6.60524088412105552835267510769, 7.13192995533801377154815530068, 8.404504717880127472182545369908