| L(s) = 1 | − 2.41·3-s − 1.82·5-s + 4·7-s + 2.82·9-s + 0.414·11-s + 6.65·13-s + 4.41·15-s − 7.65·17-s + 2·19-s − 9.65·21-s − 0.828·23-s − 1.65·25-s + 0.414·27-s − 29-s + 5.58·31-s − 0.999·33-s − 7.31·35-s − 9.65·37-s − 16.0·39-s + 1.65·41-s + 4.41·43-s − 5.17·45-s − 2.07·47-s + 9·49-s + 18.4·51-s + 7·53-s − 0.757·55-s + ⋯ |
| L(s) = 1 | − 1.39·3-s − 0.817·5-s + 1.51·7-s + 0.942·9-s + 0.124·11-s + 1.84·13-s + 1.13·15-s − 1.85·17-s + 0.458·19-s − 2.10·21-s − 0.172·23-s − 0.331·25-s + 0.0797·27-s − 0.185·29-s + 1.00·31-s − 0.174·33-s − 1.23·35-s − 1.58·37-s − 2.57·39-s + 0.258·41-s + 0.673·43-s − 0.770·45-s − 0.302·47-s + 1.28·49-s + 2.58·51-s + 0.961·53-s − 0.102·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.037547675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.037547675\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 + 1.82T + 5T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 - 0.414T + 11T^{2} \) |
| 13 | \( 1 - 6.65T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 + 9.65T + 37T^{2} \) |
| 41 | \( 1 - 1.65T + 41T^{2} \) |
| 43 | \( 1 - 4.41T + 43T^{2} \) |
| 47 | \( 1 + 2.07T + 47T^{2} \) |
| 53 | \( 1 - 7T + 53T^{2} \) |
| 59 | \( 1 + 6.48T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 + 3.17T + 83T^{2} \) |
| 89 | \( 1 - 1.65T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−9.007372830454827090878401281455, −8.425104430547843414675274448455, −7.68895818865008627869604175828, −6.68766320361862936471883108962, −6.06348173685324244072963451437, −5.11394571623767080358112439905, −4.47205584348739006362095395391, −3.69856618513812774467495535229, −1.90105453027717880607339327404, −0.76757301478220542114397743374,
0.76757301478220542114397743374, 1.90105453027717880607339327404, 3.69856618513812774467495535229, 4.47205584348739006362095395391, 5.11394571623767080358112439905, 6.06348173685324244072963451437, 6.68766320361862936471883108962, 7.68895818865008627869604175828, 8.425104430547843414675274448455, 9.007372830454827090878401281455
