| L(s) = 1 | − 2.58·2-s + 4.70·4-s + 5-s − 7-s − 6.99·8-s − 2.58·10-s + 11-s + 6.40·13-s + 2.58·14-s + 8.70·16-s + 3.70·17-s + 7.24·19-s + 4.70·20-s − 2.58·22-s + 7.33·23-s + 25-s − 16.5·26-s − 4.70·28-s + 8.10·29-s + 6.77·31-s − 8.53·32-s − 9.58·34-s − 35-s − 3.95·37-s − 18.7·38-s − 6.99·40-s + 2.31·41-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 2.35·4-s + 0.447·5-s − 0.377·7-s − 2.47·8-s − 0.818·10-s + 0.301·11-s + 1.77·13-s + 0.691·14-s + 2.17·16-s + 0.897·17-s + 1.66·19-s + 1.05·20-s − 0.551·22-s + 1.52·23-s + 0.200·25-s − 3.25·26-s − 0.888·28-s + 1.50·29-s + 1.21·31-s − 1.50·32-s − 1.64·34-s − 0.169·35-s − 0.649·37-s − 3.04·38-s − 1.10·40-s + 0.361·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.144789172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.144789172\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 13 | \( 1 - 6.40T + 13T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 - 7.24T + 19T^{2} \) |
| 23 | \( 1 - 7.33T + 23T^{2} \) |
| 29 | \( 1 - 8.10T + 29T^{2} \) |
| 31 | \( 1 - 6.77T + 31T^{2} \) |
| 37 | \( 1 + 3.95T + 37T^{2} \) |
| 41 | \( 1 - 2.31T + 41T^{2} \) |
| 43 | \( 1 + 7.65T + 43T^{2} \) |
| 47 | \( 1 + 1.45T + 47T^{2} \) |
| 53 | \( 1 + 9.10T + 53T^{2} \) |
| 59 | \( 1 - 2.92T + 59T^{2} \) |
| 61 | \( 1 + 8.42T + 61T^{2} \) |
| 67 | \( 1 + 8.72T + 67T^{2} \) |
| 71 | \( 1 - 4.40T + 71T^{2} \) |
| 73 | \( 1 - 3.40T + 73T^{2} \) |
| 79 | \( 1 - 4.40T + 79T^{2} \) |
| 83 | \( 1 - 8.96T + 83T^{2} \) |
| 89 | \( 1 + 0.206T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−8.575183753654952319036162087682, −8.157432515668550748528476775589, −7.23535576065560048380676879006, −6.55664772780740128837539043403, −6.03038166006685800855290018484, −4.99964411981494983740321986447, −3.37363334189694652171519174510, −2.88548481491767205819014834690, −1.36175558096588281439447502201, −1.00644691164110150661023317506,
1.00644691164110150661023317506, 1.36175558096588281439447502201, 2.88548481491767205819014834690, 3.37363334189694652171519174510, 4.99964411981494983740321986447, 6.03038166006685800855290018484, 6.55664772780740128837539043403, 7.23535576065560048380676879006, 8.157432515668550748528476775589, 8.575183753654952319036162087682
