| L(s) = 1 | − 3-s + 0.957·5-s − 0.898·7-s + 9-s − 0.936·11-s + 1.86·13-s − 0.957·15-s + 6.88·17-s + 0.644·19-s + 0.898·21-s − 1.95·23-s − 4.08·25-s − 27-s − 0.602·29-s + 3.41·31-s + 0.936·33-s − 0.860·35-s + 11.6·37-s − 1.86·39-s + 0.378·41-s − 8.52·43-s + 0.957·45-s − 1.26·47-s − 6.19·49-s − 6.88·51-s − 9.54·53-s − 0.896·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.428·5-s − 0.339·7-s + 0.333·9-s − 0.282·11-s + 0.517·13-s − 0.247·15-s + 1.67·17-s + 0.147·19-s + 0.196·21-s − 0.408·23-s − 0.816·25-s − 0.192·27-s − 0.111·29-s + 0.613·31-s + 0.163·33-s − 0.145·35-s + 1.92·37-s − 0.298·39-s + 0.0591·41-s − 1.30·43-s + 0.142·45-s − 0.184·47-s − 0.884·49-s − 0.964·51-s − 1.31·53-s − 0.120·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.639256205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.639256205\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 0.957T + 5T^{2} \) |
| 7 | \( 1 + 0.898T + 7T^{2} \) |
| 11 | \( 1 + 0.936T + 11T^{2} \) |
| 13 | \( 1 - 1.86T + 13T^{2} \) |
| 17 | \( 1 - 6.88T + 17T^{2} \) |
| 19 | \( 1 - 0.644T + 19T^{2} \) |
| 23 | \( 1 + 1.95T + 23T^{2} \) |
| 29 | \( 1 + 0.602T + 29T^{2} \) |
| 31 | \( 1 - 3.41T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 - 0.378T + 41T^{2} \) |
| 43 | \( 1 + 8.52T + 43T^{2} \) |
| 47 | \( 1 + 1.26T + 47T^{2} \) |
| 53 | \( 1 + 9.54T + 53T^{2} \) |
| 59 | \( 1 + 1.15T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 7.35T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 0.169T + 73T^{2} \) |
| 79 | \( 1 + 3.88T + 79T^{2} \) |
| 83 | \( 1 + 3.18T + 83T^{2} \) |
| 89 | \( 1 - 6.13T + 89T^{2} \) |
| 97 | \( 1 - 6.69T + 97T^{2} \) |
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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
−8.172786448966160534510779832037, −7.88640882836409629792445918404, −6.81838719166190874834474809630, −6.12564171474546319447826151018, −5.61071820139526338808146289161, −4.83226259174891977202507898204, −3.81151695335896675534724818327, −3.03000235823453059219726460034, −1.84516666880677146367710498543, −0.77323578820868299571850019550,
0.77323578820868299571850019550, 1.84516666880677146367710498543, 3.03000235823453059219726460034, 3.81151695335896675534724818327, 4.83226259174891977202507898204, 5.61071820139526338808146289161, 6.12564171474546319447826151018, 6.81838719166190874834474809630, 7.88640882836409629792445918404, 8.172786448966160534510779832037
