L(s) = 1 | + 2-s + 4-s − 1.15i·5-s − 7-s + 8-s − 1.15i·10-s + 2.90i·11-s − 1.38i·13-s − 14-s + 16-s + 2.90i·17-s + (3.79 − 2.14i)19-s − 1.15i·20-s + 2.90i·22-s − 7.14i·23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.516i·5-s − 0.377·7-s + 0.353·8-s − 0.365i·10-s + 0.876i·11-s − 0.383i·13-s − 0.267·14-s + 0.250·16-s + 0.704i·17-s + (0.870 − 0.492i)19-s − 0.258i·20-s + 0.619i·22-s − 1.49i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.835588674\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.835588674\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + (-3.79 + 2.14i)T \) |
good | 5 | \( 1 + 1.15iT - 5T^{2} \) |
| 11 | \( 1 - 2.90iT - 11T^{2} \) |
| 13 | \( 1 + 1.38iT - 13T^{2} \) |
| 17 | \( 1 - 2.90iT - 17T^{2} \) |
| 23 | \( 1 + 7.14iT - 23T^{2} \) |
| 29 | \( 1 - 4.66T + 29T^{2} \) |
| 31 | \( 1 - 4.06iT - 31T^{2} \) |
| 37 | \( 1 + 6.55iT - 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 2.04T + 43T^{2} \) |
| 47 | \( 1 - 2.76iT - 47T^{2} \) |
| 53 | \( 1 - 7.39T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8.29T + 61T^{2} \) |
| 67 | \( 1 + 9.91iT - 67T^{2} \) |
| 71 | \( 1 - 6.06T + 71T^{2} \) |
| 73 | \( 1 - 9.37T + 73T^{2} \) |
| 79 | \( 1 + 12.1iT - 79T^{2} \) |
| 83 | \( 1 - 14.5iT - 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 + 3.92iT - 97T^{2} \) |
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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
−8.909661298522280591003362705532, −8.095117645073017626774218191251, −7.21383478895818524947975668195, −6.54860259133399355086203141675, −5.68983387172179462855958174807, −4.82721178114342681805022215331, −4.26866435678135092271891355540, −3.15752282720635513631025011471, −2.28363232779957446998454783609, −0.920840266840014272614298634841,
1.11637436356076120596049565984, 2.61183987231393619658795783646, 3.25145137255268590413411492910, 4.07796919582929217715903036978, 5.17685715011732312125880056534, 5.84327013359402077558600374632, 6.62905579628947495032969003137, 7.32121696175275359855389653386, 8.079749706229315404423197851205, 9.115600741773889567060854219895