| L(s) = 1 | − 1.30·3-s − 2.30·5-s + 7-s − 1.30·9-s + 6.60·13-s + 3·15-s + 17-s + 6.60·19-s − 1.30·21-s + 0.302·25-s + 5.60·27-s + 4.30·31-s − 2.30·35-s − 2.60·37-s − 8.60·39-s + 3.90·41-s + 7.30·43-s + 3.00·45-s + 4.60·47-s + 49-s − 1.30·51-s − 3.69·53-s − 8.60·57-s + 9.21·59-s − 7.90·61-s − 1.30·63-s − 15.2·65-s + ⋯ |
| L(s) = 1 | − 0.752·3-s − 1.02·5-s + 0.377·7-s − 0.434·9-s + 1.83·13-s + 0.774·15-s + 0.242·17-s + 1.51·19-s − 0.284·21-s + 0.0605·25-s + 1.07·27-s + 0.772·31-s − 0.389·35-s − 0.428·37-s − 1.37·39-s + 0.610·41-s + 1.11·43-s + 0.447·45-s + 0.671·47-s + 0.142·49-s − 0.182·51-s − 0.507·53-s − 1.13·57-s + 1.19·59-s − 1.01·61-s − 0.164·63-s − 1.88·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9646967605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9646967605\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 1.30T + 3T^{2} \) |
| 5 | \( 1 + 2.30T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6.60T + 13T^{2} \) |
| 19 | \( 1 - 6.60T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4.30T + 31T^{2} \) |
| 37 | \( 1 + 2.60T + 37T^{2} \) |
| 41 | \( 1 - 3.90T + 41T^{2} \) |
| 43 | \( 1 - 7.30T + 43T^{2} \) |
| 47 | \( 1 - 4.60T + 47T^{2} \) |
| 53 | \( 1 + 3.69T + 53T^{2} \) |
| 59 | \( 1 - 9.21T + 59T^{2} \) |
| 61 | \( 1 + 7.90T + 61T^{2} \) |
| 67 | \( 1 + 1.69T + 67T^{2} \) |
| 71 | \( 1 + 7.81T + 71T^{2} \) |
| 73 | \( 1 + 7.90T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 6.30T + 97T^{2} \) |
| 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
−11.24986295402446936010539144501, −10.43607854978295670669882775801, −9.071850479308157264512086596784, −8.231771090916395393112669749530, −7.43551211452122480267084848683, −6.20598451011036104026715758592, −5.42905692114121948332408135573, −4.21060193311728770651264991384, −3.18570536413638594638619717691, −0.986031096073677374504110326231,
0.986031096073677374504110326231, 3.18570536413638594638619717691, 4.21060193311728770651264991384, 5.42905692114121948332408135573, 6.20598451011036104026715758592, 7.43551211452122480267084848683, 8.231771090916395393112669749530, 9.071850479308157264512086596784, 10.43607854978295670669882775801, 11.24986295402446936010539144501
