| L(s) = 1 | + 131.·2-s − 395.·3-s + 9.05e3·4-s − 3.44e4·5-s − 5.18e4·6-s − 3.74e5·7-s + 1.13e5·8-s − 1.43e6·9-s − 4.52e6·10-s − 6.52e6·11-s − 3.57e6·12-s − 4.92e7·14-s + 1.36e7·15-s − 5.92e7·16-s − 1.13e8·17-s − 1.88e8·18-s + 7.26e7·19-s − 3.11e8·20-s + 1.48e8·21-s − 8.56e8·22-s − 2.61e8·23-s − 4.47e7·24-s − 3.50e7·25-s + 1.19e9·27-s − 3.39e9·28-s − 4.36e9·29-s + 1.78e9·30-s + ⋯ |
| L(s) = 1 | + 1.45·2-s − 0.312·3-s + 1.10·4-s − 0.985·5-s − 0.454·6-s − 1.20·7-s + 0.152·8-s − 0.902·9-s − 1.43·10-s − 1.11·11-s − 0.345·12-s − 1.74·14-s + 0.308·15-s − 0.883·16-s − 1.13·17-s − 1.30·18-s + 0.354·19-s − 1.08·20-s + 0.376·21-s − 1.61·22-s − 0.368·23-s − 0.0478·24-s − 0.0286·25-s + 0.595·27-s − 1.33·28-s − 1.36·29-s + 0.447·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.01387942232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01387942232\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 - 131.T + 8.19e3T^{2} \) |
| 3 | \( 1 + 395.T + 1.59e6T^{2} \) |
| 5 | \( 1 + 3.44e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 3.74e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 6.52e6T + 3.45e13T^{2} \) |
| 17 | \( 1 + 1.13e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 7.26e7T + 4.20e16T^{2} \) |
| 23 | \( 1 + 2.61e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 4.36e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 2.46e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 1.21e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 3.26e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 1.89e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 1.30e11T + 5.46e21T^{2} \) |
| 53 | \( 1 - 9.35e10T + 2.60e22T^{2} \) |
| 59 | \( 1 + 3.85e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 1.30e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 1.00e12T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.79e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.44e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 1.00e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 4.32e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 9.30e11T + 2.19e25T^{2} \) |
| 97 | \( 1 + 9.20e12T + 6.73e25T^{2} \) |
| 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
−10.92689624429600049114377423255, −9.479257281376364827982991446584, −8.227570328419461495025702055415, −7.00029107596289665827190054109, −6.04616715692544009098137543754, −5.21326613686779499380535164768, −4.09948927331430474291821814051, −3.24727473059275222639714599179, −2.45131152937887231227790875438, −0.03711744982599318284916424051,
0.03711744982599318284916424051, 2.45131152937887231227790875438, 3.24727473059275222639714599179, 4.09948927331430474291821814051, 5.21326613686779499380535164768, 6.04616715692544009098137543754, 7.00029107596289665827190054109, 8.227570328419461495025702055415, 9.479257281376364827982991446584, 10.92689624429600049114377423255
