| L(s) = 1 | − 5.09e3i·2-s − 4.89e4i·3-s − 1.75e7·4-s − 2.49e8·6-s + 5.17e9i·7-s + 4.68e10i·8-s + 9.17e10·9-s + 6.04e11·11-s + 8.61e11i·12-s + 7.96e12i·13-s + 2.63e13·14-s + 9.13e13·16-s − 1.98e13i·17-s − 4.67e14i·18-s − 6.27e14·19-s + ⋯ |
| L(s) = 1 | − 1.75i·2-s − 0.159i·3-s − 2.09·4-s − 0.280·6-s + 0.988i·7-s + 1.92i·8-s + 0.974·9-s + 0.638·11-s + 0.334i·12-s + 1.23i·13-s + 1.73·14-s + 1.29·16-s − 0.140i·17-s − 1.71i·18-s − 1.23·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(0.7355007488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7355007488\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 5.09e3iT - 8.38e6T^{2} \) |
| 3 | \( 1 + 4.89e4iT - 9.41e10T^{2} \) |
| 7 | \( 1 - 5.17e9iT - 2.73e19T^{2} \) |
| 11 | \( 1 - 6.04e11T + 8.95e23T^{2} \) |
| 13 | \( 1 - 7.96e12iT - 4.17e25T^{2} \) |
| 17 | \( 1 + 1.98e13iT - 1.99e28T^{2} \) |
| 19 | \( 1 + 6.27e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 4.55e15iT - 2.08e31T^{2} \) |
| 29 | \( 1 + 4.14e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 1.35e15T + 2.00e34T^{2} \) |
| 37 | \( 1 + 3.41e17iT - 1.17e36T^{2} \) |
| 41 | \( 1 + 3.69e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 1.96e18iT - 3.71e37T^{2} \) |
| 47 | \( 1 + 2.44e19iT - 2.87e38T^{2} \) |
| 53 | \( 1 + 6.39e19iT - 4.55e39T^{2} \) |
| 59 | \( 1 + 2.81e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 4.67e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 2.77e20iT - 9.99e41T^{2} \) |
| 71 | \( 1 - 2.29e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 4.56e21iT - 7.18e42T^{2} \) |
| 79 | \( 1 - 3.99e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 1.45e22iT - 1.37e44T^{2} \) |
| 89 | \( 1 + 1.80e21T + 6.85e44T^{2} \) |
| 97 | \( 1 + 8.25e22iT - 4.96e45T^{2} \) |
| 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
−11.90798639298045949157849407846, −10.72136411284021317709412300175, −9.500350651896996109976146949686, −8.675940200113596350051257653839, −6.59692827374557511670932271893, −4.71688623555733539935701326234, −3.75954597036496812831918003610, −2.22195719309269315436793483941, −1.65395163169862815011536875156, −0.18081500566196787160030800245,
1.17569854228856734047869809353, 3.76006257268661187037076420425, 4.71116350462491081795460595498, 6.08444057633189353110923563882, 7.15666112050234910371011308657, 7.994430451691739993301766211689, 9.403361369385964169617375125815, 10.60137634230450134253677876668, 12.81139281792100282032253951489, 13.72075960592634082559269186185
