L(s) = 1 | − i·2-s − 4-s + (0.707 + 2.12i)5-s − 2i·7-s + i·8-s + (2.12 − 0.707i)10-s + 4.24·11-s + 0.828i·13-s − 2·14-s + 16-s − 6.82i·17-s − 6.24·19-s + (−0.707 − 2.12i)20-s − 4.24i·22-s − i·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.316 + 0.948i)5-s − 0.755i·7-s + 0.353i·8-s + (0.670 − 0.223i)10-s + 1.27·11-s + 0.229i·13-s − 0.534·14-s + 0.250·16-s − 1.65i·17-s − 1.43·19-s + (−0.158 − 0.474i)20-s − 0.904i·22-s − 0.208i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.795406418\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.795406418\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 - 2.12i)T \) |
| 23 | \( 1 + iT \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 - 0.828iT - 13T^{2} \) |
| 17 | \( 1 + 6.82iT - 17T^{2} \) |
| 19 | \( 1 + 6.24T + 19T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 0.585iT - 37T^{2} \) |
| 41 | \( 1 - 6.82T + 41T^{2} \) |
| 43 | \( 1 + 10.2iT - 43T^{2} \) |
| 47 | \( 1 - 0.828iT - 47T^{2} \) |
| 53 | \( 1 - 10.5iT - 53T^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 + 0.585T + 61T^{2} \) |
| 67 | \( 1 + 3.41iT - 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 7.65iT - 73T^{2} \) |
| 79 | \( 1 - 3.65T + 79T^{2} \) |
| 83 | \( 1 - 1.41iT - 83T^{2} \) |
| 89 | \( 1 - 9.17T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−9.225760228984908953303623623949, −8.336155582719007915098940068401, −7.18956477835526013588411194887, −6.75374571470554442842860986713, −5.88973481229376519254923396825, −4.57958555560960042480570660286, −3.99570044710681063594551256301, −2.97318003433432858218085999231, −2.10918113468811383981700111709, −0.76895750318965256275921715865,
1.12210723840349955897370210286, 2.25360677275411575432899391000, 3.85237141153582884596001411539, 4.44904087140195915873529428601, 5.44112444665774667049133513137, 6.24596870132349772215646284552, 6.56520799814592376457942129148, 8.073354133070733301045638621887, 8.410849774560288694443747320913, 9.067041115151478384263832741985