| L(s) = 1 | + 2i·7-s + 34·11-s + 68i·13-s + 38i·17-s − 4·19-s + 152i·23-s − 46·29-s − 260·31-s − 312i·37-s − 48·41-s + 200i·43-s − 104i·47-s + 339·49-s − 414i·53-s − 2·59-s + ⋯ |
| L(s) = 1 | + 0.107i·7-s + 0.931·11-s + 1.45i·13-s + 0.542i·17-s − 0.0482·19-s + 1.37i·23-s − 0.294·29-s − 1.50·31-s − 1.38i·37-s − 0.182·41-s + 0.709i·43-s − 0.322i·47-s + 0.988·49-s − 1.07i·53-s − 0.00441·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.046462934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.046462934\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2iT - 343T^{2} \) |
| 11 | \( 1 - 34T + 1.33e3T^{2} \) |
| 13 | \( 1 - 68iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 38iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 152iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 46T + 2.43e4T^{2} \) |
| 31 | \( 1 + 260T + 2.97e4T^{2} \) |
| 37 | \( 1 + 312iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 48T + 6.89e4T^{2} \) |
| 43 | \( 1 - 200iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 104iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 414iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 38T + 2.26e5T^{2} \) |
| 67 | \( 1 + 244iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 708T + 3.57e5T^{2} \) |
| 73 | \( 1 - 378iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 852T + 4.93e5T^{2} \) |
| 83 | \( 1 - 844iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 514iT - 9.12e5T^{2} \) |
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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
−9.231134069541397774704516103743, −8.667573517064084180721313762779, −7.53430917894524578528311617551, −6.92358572374441398203106559622, −6.08051131642513574113777120786, −5.26542185060633562009835004908, −4.08997769728353709825916493531, −3.62967561829915436367724935480, −2.12750214552476684014645198399, −1.38269724433825071877707797595,
0.22296485320069421988138576863, 1.27361163784536643386691852790, 2.57942630530824529166706273685, 3.48038113598410701114783602035, 4.43042803515601074410462941577, 5.36283367019598991585035087968, 6.16499611670839649292858219902, 7.03133855205774763262702946724, 7.76673501808797670196518565726, 8.654230938907825612791530276561
