| L(s) = 1 | − 6.77e3·2-s + 4.09e5·3-s + 1.23e7·4-s + 2.95e8·5-s − 2.77e9·6-s + 1.43e11·8-s − 6.79e11·9-s − 2.00e12·10-s + 2.31e12·11-s + 5.04e12·12-s − 1.47e14·13-s + 1.21e14·15-s − 1.38e15·16-s − 3.82e15·17-s + 4.60e15·18-s − 4.18e15·19-s + 3.64e15·20-s − 1.57e16·22-s − 6.35e16·23-s + 5.89e16·24-s − 2.10e17·25-s + 9.98e17·26-s − 6.25e17·27-s − 2.10e18·29-s − 8.20e17·30-s − 1.11e18·31-s + 4.57e18·32-s + ⋯ |
| L(s) = 1 | − 1.16·2-s + 0.444·3-s + 0.367·4-s + 0.542·5-s − 0.520·6-s + 0.739·8-s − 0.802·9-s − 0.633·10-s + 0.222·11-s + 0.163·12-s − 1.75·13-s + 0.241·15-s − 1.23·16-s − 1.59·17-s + 0.937·18-s − 0.433·19-s + 0.199·20-s − 0.260·22-s − 0.604·23-s + 0.329·24-s − 0.706·25-s + 2.05·26-s − 0.801·27-s − 1.10·29-s − 0.282·30-s − 0.255·31-s + 0.701·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(0.2377779264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2377779264\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 6.77e3T + 3.35e7T^{2} \) |
| 3 | \( 1 - 4.09e5T + 8.47e11T^{2} \) |
| 5 | \( 1 - 2.95e8T + 2.98e17T^{2} \) |
| 11 | \( 1 - 2.31e12T + 1.08e26T^{2} \) |
| 13 | \( 1 + 1.47e14T + 7.05e27T^{2} \) |
| 17 | \( 1 + 3.82e15T + 5.77e30T^{2} \) |
| 19 | \( 1 + 4.18e15T + 9.30e31T^{2} \) |
| 23 | \( 1 + 6.35e16T + 1.10e34T^{2} \) |
| 29 | \( 1 + 2.10e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 1.11e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 5.33e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 3.82e19T + 2.08e40T^{2} \) |
| 43 | \( 1 - 2.38e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 6.40e20T + 6.34e41T^{2} \) |
| 53 | \( 1 - 2.95e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 2.38e22T + 1.86e44T^{2} \) |
| 61 | \( 1 - 3.86e21T + 4.29e44T^{2} \) |
| 67 | \( 1 + 3.90e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 2.27e23T + 1.91e46T^{2} \) |
| 73 | \( 1 - 1.25e23T + 3.82e46T^{2} \) |
| 79 | \( 1 - 7.68e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 5.33e22T + 9.48e47T^{2} \) |
| 89 | \( 1 - 1.73e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 8.77e24T + 4.66e49T^{2} \) |
| 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
−10.53474601203705806430541928967, −9.515493407366186972233262140947, −8.915037829610714817790022045223, −7.86201964213313479412350817143, −6.83162036756079923443525485778, −5.37515622394491544496284371933, −4.11099599292289029167117769235, −2.39276682751047639157152613419, −1.90825498174236848535965464576, −0.22558844903335827426187997272,
0.22558844903335827426187997272, 1.90825498174236848535965464576, 2.39276682751047639157152613419, 4.11099599292289029167117769235, 5.37515622394491544496284371933, 6.83162036756079923443525485778, 7.86201964213313479412350817143, 8.915037829610714817790022045223, 9.515493407366186972233262140947, 10.53474601203705806430541928967
