| L(s) = 1 | − 27.6·2-s − 44.2·3-s + 254.·4-s + 2.46e3·5-s + 1.22e3·6-s − 3.58e3·7-s + 7.14e3·8-s − 1.77e4·9-s − 6.82e4·10-s + 7.92e4·11-s − 1.12e4·12-s + 9.90e4·14-s − 1.09e5·15-s − 3.27e5·16-s + 8.42e3·17-s + 4.90e5·18-s + 4.34e5·19-s + 6.26e5·20-s + 1.58e5·21-s − 2.19e6·22-s + 1.39e6·23-s − 3.15e5·24-s + 4.13e6·25-s + 1.65e6·27-s − 9.09e5·28-s + 2.73e6·29-s + 3.02e6·30-s + ⋯ |
| L(s) = 1 | − 1.22·2-s − 0.315·3-s + 0.496·4-s + 1.76·5-s + 0.385·6-s − 0.563·7-s + 0.616·8-s − 0.900·9-s − 2.15·10-s + 1.63·11-s − 0.156·12-s + 0.689·14-s − 0.556·15-s − 1.24·16-s + 0.0244·17-s + 1.10·18-s + 0.765·19-s + 0.875·20-s + 0.177·21-s − 1.99·22-s + 1.03·23-s − 0.194·24-s + 2.11·25-s + 0.599·27-s − 0.279·28-s + 0.718·29-s + 0.680·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.438298620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.438298620\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 27.6T + 512T^{2} \) |
| 3 | \( 1 + 44.2T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.46e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.58e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.92e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 8.42e3T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.34e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.39e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.73e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.77e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.57e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.12e5T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.68e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.59e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.83e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.46e5T + 8.66e15T^{2} \) |
| 61 | \( 1 - 4.42e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.65e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.69e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 8.20e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.10e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.20e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.97e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.57e8T + 7.60e17T^{2} \) |
| 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
−10.77914774318133145178242941410, −9.634248584266898275453383240808, −9.416491288983169513454565228762, −8.442933354220718453462375495044, −6.77491091964565548722491923360, −6.17045025266794680188597724494, −4.92184760490312925264335128014, −2.97551924203106717331472652249, −1.60780227677873792212305543978, −0.791093685801411966944518458910,
0.791093685801411966944518458910, 1.60780227677873792212305543978, 2.97551924203106717331472652249, 4.92184760490312925264335128014, 6.17045025266794680188597724494, 6.77491091964565548722491923360, 8.442933354220718453462375495044, 9.416491288983169513454565228762, 9.634248584266898275453383240808, 10.77914774318133145178242941410
