| L(s) = 1 | + 2.44i·5-s + 1.41·7-s + 3.46i·11-s + 4.89i·13-s − 4·17-s + 6.92i·19-s − 5.65·23-s − 0.999·25-s − 2.44i·29-s + 1.41·31-s + 3.46i·35-s − 4.89i·37-s + 4·41-s + 6.92i·43-s − 5.65·47-s + ⋯ |
| L(s) = 1 | + 1.09i·5-s + 0.534·7-s + 1.04i·11-s + 1.35i·13-s − 0.970·17-s + 1.58i·19-s − 1.17·23-s − 0.199·25-s − 0.454i·29-s + 0.254·31-s + 0.585i·35-s − 0.805i·37-s + 0.624·41-s + 1.05i·43-s − 0.825·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.186302007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.186302007\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.44iT - 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 2.44iT - 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 4.89iT - 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 6.92iT - 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 + 7.34iT - 53T^{2} \) |
| 59 | \( 1 + 13.8iT - 59T^{2} \) |
| 61 | \( 1 - 4.89iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 7.07T + 79T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.585703330973709098046560826675, −7.82884214227731826317168477297, −7.24948599460256826875784460904, −6.46262638260214270195637021796, −6.05827054984091522424873383665, −4.77199332909725140760004883113, −4.24195189137638357927175986568, −3.40679929970281412604269734086, −2.13087961339025889368741440556, −1.82457177871839811916768259937,
0.32870896004056829805300434544, 1.18683181396292337157923041747, 2.43673715634598746931364784728, 3.31200740707122780723997204234, 4.42281672709537907004245091091, 4.93704060832565329689298276663, 5.66464964025312898423679452786, 6.37695971432966126054013877521, 7.39345182180390141058530174372, 8.128342519525691993565094108678
