| L(s) = 1 | + 2.09·3-s − 1.01·5-s − 2.31·7-s + 1.40·9-s − 11-s − 0.529·13-s − 2.12·15-s − 1.32·17-s − 2.86·19-s − 4.86·21-s + 2.76·23-s − 3.97·25-s − 3.33·27-s − 1.36·29-s + 5.39·31-s − 2.09·33-s + 2.34·35-s + 8.30·37-s − 1.11·39-s + 7.64·41-s + 12.4·43-s − 1.42·45-s − 3.66·47-s − 1.62·49-s − 2.78·51-s + 53-s + 1.01·55-s + ⋯ |
| L(s) = 1 | + 1.21·3-s − 0.452·5-s − 0.876·7-s + 0.469·9-s − 0.301·11-s − 0.146·13-s − 0.548·15-s − 0.321·17-s − 0.657·19-s − 1.06·21-s + 0.576·23-s − 0.795·25-s − 0.642·27-s − 0.253·29-s + 0.969·31-s − 0.365·33-s + 0.396·35-s + 1.36·37-s − 0.178·39-s + 1.19·41-s + 1.90·43-s − 0.212·45-s − 0.535·47-s − 0.231·49-s − 0.389·51-s + 0.137·53-s + 0.136·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.049511768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.049511768\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 - 2.09T + 3T^{2} \) |
| 5 | \( 1 + 1.01T + 5T^{2} \) |
| 7 | \( 1 + 2.31T + 7T^{2} \) |
| 13 | \( 1 + 0.529T + 13T^{2} \) |
| 17 | \( 1 + 1.32T + 17T^{2} \) |
| 19 | \( 1 + 2.86T + 19T^{2} \) |
| 23 | \( 1 - 2.76T + 23T^{2} \) |
| 29 | \( 1 + 1.36T + 29T^{2} \) |
| 31 | \( 1 - 5.39T + 31T^{2} \) |
| 37 | \( 1 - 8.30T + 37T^{2} \) |
| 41 | \( 1 - 7.64T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 + 3.66T + 47T^{2} \) |
| 59 | \( 1 + 0.571T + 59T^{2} \) |
| 61 | \( 1 + 4.93T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 2.21T + 71T^{2} \) |
| 73 | \( 1 - 3.46T + 73T^{2} \) |
| 79 | \( 1 + 1.92T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 2.71T + 89T^{2} \) |
| 97 | \( 1 - 0.367T + 97T^{2} \) |
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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
−7.69762281139901718336606447036, −7.34143020172011339238695490711, −6.28858220878131645691497212393, −5.89664310010716365166863319945, −4.64576422604720447195008549792, −4.10658956443277937386318490756, −3.29512471118096772770755019428, −2.72606571833783531595077726531, −2.05560911867416497702870551640, −0.61452761889362409037241623561,
0.61452761889362409037241623561, 2.05560911867416497702870551640, 2.72606571833783531595077726531, 3.29512471118096772770755019428, 4.10658956443277937386318490756, 4.64576422604720447195008549792, 5.89664310010716365166863319945, 6.28858220878131645691497212393, 7.34143020172011339238695490711, 7.69762281139901718336606447036
