| L(s) = 1 | + (−1.5 − 2.59i)3-s + (−7 + 12.1i)5-s + (−4.5 + 7.79i)9-s + (−2 − 3.46i)11-s + 54·13-s + 42·15-s + (7 + 12.1i)17-s + (−46 + 79.6i)19-s + (76 − 131. i)23-s + (−35.5 − 61.4i)25-s + 27·27-s − 106·29-s + (72 + 124. i)31-s + (−6 + 10.3i)33-s + (−79 + 136. i)37-s + ⋯ |
| L(s) = 1 | + (−0.288 − 0.499i)3-s + (−0.626 + 1.08i)5-s + (−0.166 + 0.288i)9-s + (−0.0548 − 0.0949i)11-s + 1.15·13-s + 0.722·15-s + (0.0998 + 0.172i)17-s + (−0.555 + 0.962i)19-s + (0.689 − 1.19i)23-s + (−0.284 − 0.491i)25-s + 0.192·27-s − 0.678·29-s + (0.417 + 0.722i)31-s + (−0.0316 + 0.0548i)33-s + (−0.351 + 0.607i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2901319911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2901319911\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 + 2.59i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (7 - 12.1i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (2 + 3.46i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 54T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-7 - 12.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (46 - 79.6i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-76 + 131. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 106T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-72 - 124. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (79 - 136. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 390T + 6.89e4T^{2} \) |
| 43 | \( 1 + 508T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-264 + 457. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (303 + 524. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-182 - 315. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (339 - 587. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (422 + 730. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 8T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-211 - 365. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (192 - 332. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 548T + 5.71e5T^{2} \) |
| 89 | \( 1 + (597 - 1.03e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.50e3T + 9.12e5T^{2} \) |
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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
−10.71007958256415144953456624160, −10.22113591129320116489378126048, −8.629662634215789331548687531877, −8.109095036545003572351853855640, −6.87520399321193183965056400294, −6.52315648564995398553627961534, −5.32396337660189184897287211027, −3.89745433927964674163016213857, −3.02702136813274256252712982239, −1.56156187896232319924575176752,
0.093945551802059977620418330843, 1.38160132855859268099628637257, 3.27192605291385878575052164001, 4.28975681911657657082182059729, 5.05862910910488125454326118382, 6.04700679747593840765199914588, 7.24204085428413570417077202273, 8.327907883784595903889668106389, 8.946876767163782537914800335277, 9.747689147172184987445118173993
