| L(s) = 1 | − 3-s + (−2 − i)5-s + 9-s + 2i·11-s + 2·13-s + (2 + i)15-s − 2i·17-s − 4i·19-s + (3 + 4i)25-s − 27-s + 2i·29-s − 2·31-s − 2i·33-s − 6·37-s − 2·39-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (−0.894 − 0.447i)5-s + 0.333·9-s + 0.603i·11-s + 0.554·13-s + (0.516 + 0.258i)15-s − 0.485i·17-s − 0.917i·19-s + (0.600 + 0.800i)25-s − 0.192·27-s + 0.371i·29-s − 0.359·31-s − 0.348i·33-s − 0.986·37-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9597921210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9597921210\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (2 + i)T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 - 10iT - 59T^{2} \) |
| 61 | \( 1 + 10iT - 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−8.301135734302670680208046400355, −7.50572255743929007037414682692, −6.98386487209671710766031599212, −6.14197394679537131816096484584, −5.16347497495768132013163815834, −4.63976968736295292623419476655, −3.85558900432855960213260413017, −2.88809972389325135170118993022, −1.54366535092490887031127154544, −0.42541889975887720567774235352,
0.844106947681959417305958526247, 2.15262796400271033643395040294, 3.52316725664778170002318278956, 3.80100068349222264527178856770, 4.89450300098607313410359669191, 5.75473836946086729294325797202, 6.41081540659063437805373555171, 7.10790709231841967364674308538, 7.973577543265624712970010402681, 8.413844143971871309213186901728
