| L(s) = 1 | − 2.08·3-s + 0.209·5-s − 7-s + 1.35·9-s − 6.03·11-s + 3.67·13-s − 0.437·15-s − 5.37·17-s + 3.54·19-s + 2.08·21-s + 1.30·23-s − 4.95·25-s + 3.43·27-s − 8.00·29-s + 0.384·31-s + 12.6·33-s − 0.209·35-s − 3.68·37-s − 7.66·39-s − 41-s − 0.824·43-s + 0.284·45-s − 5.11·47-s + 49-s + 11.2·51-s + 1.53·53-s − 1.26·55-s + ⋯ |
| L(s) = 1 | − 1.20·3-s + 0.0937·5-s − 0.377·7-s + 0.452·9-s − 1.82·11-s + 1.01·13-s − 0.112·15-s − 1.30·17-s + 0.812·19-s + 0.455·21-s + 0.271·23-s − 0.991·25-s + 0.660·27-s − 1.48·29-s + 0.0690·31-s + 2.19·33-s − 0.0354·35-s − 0.606·37-s − 1.22·39-s − 0.156·41-s − 0.125·43-s + 0.0424·45-s − 0.746·47-s + 0.142·49-s + 1.56·51-s + 0.210·53-s − 0.170·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5001765682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5001765682\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 2.08T + 3T^{2} \) |
| 5 | \( 1 - 0.209T + 5T^{2} \) |
| 11 | \( 1 + 6.03T + 11T^{2} \) |
| 13 | \( 1 - 3.67T + 13T^{2} \) |
| 17 | \( 1 + 5.37T + 17T^{2} \) |
| 19 | \( 1 - 3.54T + 19T^{2} \) |
| 23 | \( 1 - 1.30T + 23T^{2} \) |
| 29 | \( 1 + 8.00T + 29T^{2} \) |
| 31 | \( 1 - 0.384T + 31T^{2} \) |
| 37 | \( 1 + 3.68T + 37T^{2} \) |
| 43 | \( 1 + 0.824T + 43T^{2} \) |
| 47 | \( 1 + 5.11T + 47T^{2} \) |
| 53 | \( 1 - 1.53T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 9.36T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 - 7.77T + 73T^{2} \) |
| 79 | \( 1 - 6.04T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 0.520T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−8.249694054860245499431113772792, −7.51824732632437233797966475999, −6.72191977143176221168578307912, −6.01461729637817961247381453840, −5.43466498037387321812522663935, −4.90610261176681705857037393633, −3.83813422468472842032526996144, −2.89946956215682661719598294025, −1.84444243284089312831468106322, −0.40154648597531689929722493213,
0.40154648597531689929722493213, 1.84444243284089312831468106322, 2.89946956215682661719598294025, 3.83813422468472842032526996144, 4.90610261176681705857037393633, 5.43466498037387321812522663935, 6.01461729637817961247381453840, 6.72191977143176221168578307912, 7.51824732632437233797966475999, 8.249694054860245499431113772792
