| L(s) = 1 | − 2-s + 4-s − 3.17·5-s − 3.35·7-s − 8-s + 3.17·10-s − 2.44·11-s + 5.10·13-s + 3.35·14-s + 16-s − 3.15·17-s − 3.69·19-s − 3.17·20-s + 2.44·22-s + 5.08·25-s − 5.10·26-s − 3.35·28-s − 9.59·29-s + 3.46·31-s − 32-s + 3.15·34-s + 10.6·35-s + 3.35·37-s + 3.69·38-s + 3.17·40-s + 1.82·41-s − 10.9·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.41·5-s − 1.26·7-s − 0.353·8-s + 1.00·10-s − 0.738·11-s + 1.41·13-s + 0.897·14-s + 0.250·16-s − 0.764·17-s − 0.847·19-s − 0.709·20-s + 0.522·22-s + 1.01·25-s − 1.00·26-s − 0.634·28-s − 1.78·29-s + 0.622·31-s − 0.176·32-s + 0.540·34-s + 1.80·35-s + 0.552·37-s + 0.599·38-s + 0.502·40-s + 0.284·41-s − 1.66·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08945163570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08945163570\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 3.17T + 5T^{2} \) |
| 7 | \( 1 + 3.35T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - 5.10T + 13T^{2} \) |
| 17 | \( 1 + 3.15T + 17T^{2} \) |
| 19 | \( 1 + 3.69T + 19T^{2} \) |
| 29 | \( 1 + 9.59T + 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 3.35T + 37T^{2} \) |
| 41 | \( 1 - 1.82T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 7.95T + 47T^{2} \) |
| 53 | \( 1 + 3.70T + 53T^{2} \) |
| 59 | \( 1 + 0.330T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 - 1.58T + 67T^{2} \) |
| 71 | \( 1 + 9.25T + 71T^{2} \) |
| 73 | \( 1 + 7.66T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 2.27T + 83T^{2} \) |
| 89 | \( 1 + 1.78T + 89T^{2} \) |
| 97 | \( 1 + 1.34T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.83945878116784409256492955601, −7.10531095953027690620682096544, −6.41520523647881032142711313187, −5.99961558272513678563351055389, −4.82947385387388014137399522712, −3.94710304163537119539116959988, −3.44451896577753156113562253842, −2.72916223728598444116356418232, −1.54601136079420687308164094913, −0.15734844204356683653618836233,
0.15734844204356683653618836233, 1.54601136079420687308164094913, 2.72916223728598444116356418232, 3.44451896577753156113562253842, 3.94710304163537119539116959988, 4.82947385387388014137399522712, 5.99961558272513678563351055389, 6.41520523647881032142711313187, 7.10531095953027690620682096544, 7.83945878116784409256492955601
