L(s) = 1 | + 3.46·5-s + 4.89·7-s − 5.65·11-s + 6.99·25-s + 10.3·29-s + 4.89·31-s + 16.9·35-s + 16.9·49-s − 3.46·53-s − 19.5·55-s − 11.3·59-s + 14·73-s − 27.7·77-s − 14.6·79-s + 5.65·83-s + 2·97-s − 3.46·101-s + 14.6·103-s − 11.3·107-s + ⋯ |
L(s) = 1 | + 1.54·5-s + 1.85·7-s − 1.70·11-s + 1.39·25-s + 1.92·29-s + 0.879·31-s + 2.86·35-s + 2.42·49-s − 0.475·53-s − 2.64·55-s − 1.47·59-s + 1.63·73-s − 3.15·77-s − 1.65·79-s + 0.620·83-s + 0.203·97-s − 0.344·101-s + 1.44·103-s − 1.09·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.843917847\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.843917847\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 - 4.89T + 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 5.65T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.908886141756462347045539826382, −8.187137886637984035787864004344, −7.70001242052160483655211032578, −6.56223235327614909564345711268, −5.70396727625528097832101974596, −5.02755937968520251162104889971, −4.60657202727711500611385483625, −2.81221082798490615191322599389, −2.16917135712474218619617438038, −1.20672604666220571211921319724,
1.20672604666220571211921319724, 2.16917135712474218619617438038, 2.81221082798490615191322599389, 4.60657202727711500611385483625, 5.02755937968520251162104889971, 5.70396727625528097832101974596, 6.56223235327614909564345711268, 7.70001242052160483655211032578, 8.187137886637984035787864004344, 8.908886141756462347045539826382