| L(s) = 1 | − 0.185·3-s − 4.45·7-s − 2.96·9-s − 2.64·11-s − 1.30·13-s + 3.51·17-s + 19-s + 0.826·21-s − 6.52·23-s + 1.10·27-s − 5.20·29-s − 10.8·31-s + 0.490·33-s − 2.04·37-s + 0.241·39-s − 3.80·41-s + 4.77·43-s + 1.49·47-s + 12.8·49-s − 0.652·51-s − 0.225·53-s − 0.185·57-s + 2.86·59-s − 6.31·61-s + 13.2·63-s − 13.1·67-s + 1.21·69-s + ⋯ |
| L(s) = 1 | − 0.107·3-s − 1.68·7-s − 0.988·9-s − 0.797·11-s − 0.360·13-s + 0.853·17-s + 0.229·19-s + 0.180·21-s − 1.36·23-s + 0.212·27-s − 0.967·29-s − 1.94·31-s + 0.0854·33-s − 0.336·37-s + 0.0386·39-s − 0.593·41-s + 0.727·43-s + 0.217·47-s + 1.83·49-s − 0.0913·51-s − 0.0309·53-s − 0.0245·57-s + 0.372·59-s − 0.808·61-s + 1.66·63-s − 1.61·67-s + 0.145·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3146631768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3146631768\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.185T + 3T^{2} \) |
| 7 | \( 1 + 4.45T + 7T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 - 3.51T + 17T^{2} \) |
| 23 | \( 1 + 6.52T + 23T^{2} \) |
| 29 | \( 1 + 5.20T + 29T^{2} \) |
| 31 | \( 1 + 10.8T + 31T^{2} \) |
| 37 | \( 1 + 2.04T + 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 - 4.77T + 43T^{2} \) |
| 47 | \( 1 - 1.49T + 47T^{2} \) |
| 53 | \( 1 + 0.225T + 53T^{2} \) |
| 59 | \( 1 - 2.86T + 59T^{2} \) |
| 61 | \( 1 + 6.31T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 5.42T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 - 3.84T + 83T^{2} \) |
| 89 | \( 1 - 1.67T + 89T^{2} \) |
| 97 | \( 1 + 9.48T + 97T^{2} \) |
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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
−7.67165844610600580241192164451, −7.32062881934487265481624138652, −6.36213859715626272862413916851, −5.65970327426170206609950722487, −5.47507330449234330099584561247, −4.16421737353287476394759101987, −3.34926851278701748877809442053, −2.90305770531397944704712096849, −1.91347608636258522955062830231, −0.26121053918637492292257398979,
0.26121053918637492292257398979, 1.91347608636258522955062830231, 2.90305770531397944704712096849, 3.34926851278701748877809442053, 4.16421737353287476394759101987, 5.47507330449234330099584561247, 5.65970327426170206609950722487, 6.36213859715626272862413916851, 7.32062881934487265481624138652, 7.67165844610600580241192164451
