L(s) = 1 | + (−0.707 − 0.707i)2-s + (−1.64 − 0.542i)3-s + 1.00i·4-s + (−2.32 − 2.32i)5-s + (0.779 + 1.54i)6-s + (−1.76 − 1.76i)7-s + (0.707 − 0.707i)8-s + (2.41 + 1.78i)9-s + 3.28i·10-s + (−1.08 + 1.08i)11-s + (0.542 − 1.64i)12-s + (0.766 − 3.52i)13-s + 2.49i·14-s + (2.56 + 5.08i)15-s − 1.00·16-s + 5.73·17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.949 − 0.313i)3-s + 0.500i·4-s + (−1.04 − 1.04i)5-s + (0.318 + 0.631i)6-s + (−0.667 − 0.667i)7-s + (0.250 − 0.250i)8-s + (0.803 + 0.594i)9-s + 1.04i·10-s + (−0.327 + 0.327i)11-s + (0.156 − 0.474i)12-s + (0.212 − 0.977i)13-s + 0.667i·14-s + (0.662 + 1.31i)15-s − 0.250·16-s + 1.39·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.101482 - 0.362012i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.101482 - 0.362012i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.64 + 0.542i)T \) |
| 13 | \( 1 + (-0.766 + 3.52i)T \) |
good | 5 | \( 1 + (2.32 + 2.32i)T + 5iT^{2} \) |
| 7 | \( 1 + (1.76 + 1.76i)T + 7iT^{2} \) |
| 11 | \( 1 + (1.08 - 1.08i)T - 11iT^{2} \) |
| 17 | \( 1 - 5.73T + 17T^{2} \) |
| 19 | \( 1 + (-2.28 + 2.28i)T - 19iT^{2} \) |
| 23 | \( 1 + 4.65T + 23T^{2} \) |
| 29 | \( 1 + 4.65iT - 29T^{2} \) |
| 31 | \( 1 + (3.82 - 3.82i)T - 31iT^{2} \) |
| 37 | \( 1 + (3.05 + 3.05i)T + 37iT^{2} \) |
| 41 | \( 1 + (-0.410 - 0.410i)T + 41iT^{2} \) |
| 43 | \( 1 - 0.222iT - 43T^{2} \) |
| 47 | \( 1 + (-7.65 + 7.65i)T - 47iT^{2} \) |
| 53 | \( 1 - 10.9iT - 53T^{2} \) |
| 59 | \( 1 + (-4.65 + 4.65i)T - 59iT^{2} \) |
| 61 | \( 1 - 3.06T + 61T^{2} \) |
| 67 | \( 1 + (-0.533 + 0.533i)T - 67iT^{2} \) |
| 71 | \( 1 + (-5.48 - 5.48i)T + 71iT^{2} \) |
| 73 | \( 1 + (-2.28 - 2.28i)T + 73iT^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + (1.39 + 1.39i)T + 83iT^{2} \) |
| 89 | \( 1 + (-6.41 + 6.41i)T - 89iT^{2} \) |
| 97 | \( 1 + (11.5 - 11.5i)T - 97iT^{2} \) |
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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
−13.49661036847863529853983047165, −12.49770563619411992084178629694, −12.02781822539046693838792046587, −10.73038231481550192999613296103, −9.788986274779415543216189589009, −8.116250488168503621611169517627, −7.28821259400927405137656192568, −5.38877139318244346406308799767, −3.85016665786878510166097608901, −0.65791028884537277568242105239,
3.64454303557230889729777468219, 5.62711129380329807393199177493, 6.70106682838207334307959722476, 7.80855939658863933736575452399, 9.458727995877703871432481256466, 10.50437701126341711412792696544, 11.53495239857141919473010075133, 12.33369294520362025206555627566, 14.21437097418132054823622708361, 15.21559629337564665251279424210