L(s) = 1 | + 2-s + 4-s − 2.76i·5-s − 7-s + 8-s − 2.76i·10-s − 4.92i·11-s + 2.93i·13-s − 14-s + 16-s − 4.92i·17-s + (1.87 + 3.93i)19-s − 2.76i·20-s − 4.92i·22-s + 0.685i·23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.23i·5-s − 0.377·7-s + 0.353·8-s − 0.874i·10-s − 1.48i·11-s + 0.815i·13-s − 0.267·14-s + 0.250·16-s − 1.19i·17-s + (0.430 + 0.902i)19-s − 0.618i·20-s − 1.05i·22-s + 0.142i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.230593531\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.230593531\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + (-1.87 - 3.93i)T \) |
good | 5 | \( 1 + 2.76iT - 5T^{2} \) |
| 11 | \( 1 + 4.92iT - 11T^{2} \) |
| 13 | \( 1 - 2.93iT - 13T^{2} \) |
| 17 | \( 1 + 4.92iT - 17T^{2} \) |
| 23 | \( 1 - 0.685iT - 23T^{2} \) |
| 29 | \( 1 + 1.65T + 29T^{2} \) |
| 31 | \( 1 + 2.16iT - 31T^{2} \) |
| 37 | \( 1 + 9.77iT - 37T^{2} \) |
| 41 | \( 1 + 2.37T + 41T^{2} \) |
| 43 | \( 1 + 8.15T + 43T^{2} \) |
| 47 | \( 1 + 5.87iT - 47T^{2} \) |
| 53 | \( 1 + 3.47T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 4.26T + 61T^{2} \) |
| 67 | \( 1 - 6.56iT - 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 2.85T + 73T^{2} \) |
| 79 | \( 1 + 11.0iT - 79T^{2} \) |
| 83 | \( 1 + 7.07iT - 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 3.11iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.824351015288562818951703591713, −7.944864830595153287304240928355, −7.10680831565012160949238404905, −6.11707568240917756472272322071, −5.51182126470286779892676053455, −4.79665734066419506830509293800, −3.86021542219863392765482128887, −3.11454144729191855180490907330, −1.78972541793821234473924716144, −0.56585133844857920954791116083,
1.70707069931503326545000747798, 2.80256310434919691850517195661, 3.36070396690343673486438633902, 4.43173517490909818712294404443, 5.21960579078963665109053302490, 6.30244519511545275812387400938, 6.75093564321727165859030638452, 7.44818343732579309677871972674, 8.218099637314912287704626103040, 9.430755673601224234405927327170