| L(s) = 1 | − 1.06i·5-s + 2.36·7-s − 1.12i·11-s + 4.80i·13-s + 4.69i·17-s + (3.63 + 2.41i)19-s + 7.88i·23-s + 3.85·25-s − 3.53·29-s − 2.74i·31-s − 2.52i·35-s − 3.24i·37-s − 4.61·41-s − 3.12·43-s + 5.36i·47-s + ⋯ |
| L(s) = 1 | − 0.477i·5-s + 0.892·7-s − 0.340i·11-s + 1.33i·13-s + 1.13i·17-s + (0.832 + 0.553i)19-s + 1.64i·23-s + 0.771·25-s − 0.656·29-s − 0.493i·31-s − 0.426i·35-s − 0.533i·37-s − 0.720·41-s − 0.476·43-s + 0.782i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0290 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0290 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.693299002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.693299002\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-3.63 - 2.41i)T \) |
| good | 5 | \( 1 + 1.06iT - 5T^{2} \) |
| 7 | \( 1 - 2.36T + 7T^{2} \) |
| 11 | \( 1 + 1.12iT - 11T^{2} \) |
| 13 | \( 1 - 4.80iT - 13T^{2} \) |
| 17 | \( 1 - 4.69iT - 17T^{2} \) |
| 23 | \( 1 - 7.88iT - 23T^{2} \) |
| 29 | \( 1 + 3.53T + 29T^{2} \) |
| 31 | \( 1 + 2.74iT - 31T^{2} \) |
| 37 | \( 1 + 3.24iT - 37T^{2} \) |
| 41 | \( 1 + 4.61T + 41T^{2} \) |
| 43 | \( 1 + 3.12T + 43T^{2} \) |
| 47 | \( 1 - 5.36iT - 47T^{2} \) |
| 53 | \( 1 + 1.59T + 53T^{2} \) |
| 59 | \( 1 + 9.17T + 59T^{2} \) |
| 61 | \( 1 - 0.269T + 61T^{2} \) |
| 67 | \( 1 - 3.18iT - 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 7.11iT - 79T^{2} \) |
| 83 | \( 1 + 0.310iT - 83T^{2} \) |
| 89 | \( 1 + 2.18T + 89T^{2} \) |
| 97 | \( 1 - 9.96iT - 97T^{2} \) |
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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
−8.324899975451941158314896503943, −7.69317019383558837008519505821, −7.04747992847175284631737702026, −6.03979315678789318184655667149, −5.49317487499477777823507725636, −4.65540198834337035242524922175, −4.00419816195047939519407394593, −3.15873661984076601233979483179, −1.73361228157669195061680375067, −1.41160979699285205097264925293,
0.43560514261967658555842195350, 1.60383398874945382967516307431, 2.80444659998561151325320227841, 3.16908400019805846726268519961, 4.57066208763497192630638094615, 4.95444952559047556872361420654, 5.72745242405022809260237819328, 6.73156132210297987809274463031, 7.26506868332486704396357588287, 7.935030389954958105530015629131
