| L(s) = 1 | + (10 + 15.5i)7-s + 62.3i·13-s + 56·19-s + 125·25-s − 308·31-s − 110·37-s + 218. i·43-s + (−143 + 311. i)49-s − 935. i·61-s + 654. i·67-s − 374. i·73-s + 1.09e3i·79-s + (−972 + 623. i)91-s + 1.37e3i·97-s − 1.82e3·103-s + ⋯ |
| L(s) = 1 | + (0.539 + 0.841i)7-s + 1.33i·13-s + 0.676·19-s + 25-s − 1.78·31-s − 0.488·37-s + 0.773i·43-s + (−0.416 + 0.908i)49-s − 1.96i·61-s + 1.19i·67-s − 0.599i·73-s + 1.55i·79-s + (−1.11 + 0.718i)91-s + 1.43i·97-s − 1.74·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.539 - 0.841i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.589326041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.589326041\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-10 - 15.5i)T \) |
| good | 5 | \( 1 - 125T^{2} \) |
| 11 | \( 1 - 1.33e3T^{2} \) |
| 13 | \( 1 - 62.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 4.91e3T^{2} \) |
| 19 | \( 1 - 56T + 6.85e3T^{2} \) |
| 23 | \( 1 - 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 + 308T + 2.97e4T^{2} \) |
| 37 | \( 1 + 110T + 5.06e4T^{2} \) |
| 41 | \( 1 - 6.89e4T^{2} \) |
| 43 | \( 1 - 218. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 + 935. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 654. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 3.57e5T^{2} \) |
| 73 | \( 1 + 374. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.09e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.37e3iT - 9.12e5T^{2} \) |
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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
−9.620683968419515182834714275878, −9.089766753919397574205298494226, −8.317660404728319154835890706435, −7.32491544795479179008618003348, −6.51407709533958510304631430060, −5.46826273372656892088976420158, −4.74312067239316588792090292338, −3.60362076882675880035976877116, −2.37548257538859322609770394476, −1.39843738284671652695375467416,
0.40138826920514298592067485134, 1.51082465576029497844068844844, 2.96888567574405355099256999446, 3.88839052334495951499180559220, 5.01447141967731705340120003268, 5.69095066273922975371534327473, 6.99735235763779069581883866148, 7.55889972520756974716477914944, 8.401518793963394887060228009677, 9.286358369531523411463724613543
