L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + 1.00i·6-s − i·7-s + (0.707 − 0.707i)8-s + 1.00i·9-s + 1.00·10-s + 1.41·11-s + (0.707 − 0.707i)12-s + (−0.707 + 0.707i)14-s + 1.00·15-s − 1.00·16-s + (0.707 − 0.707i)18-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + 1.00i·6-s − i·7-s + (0.707 − 0.707i)8-s + 1.00i·9-s + 1.00·10-s + 1.41·11-s + (0.707 − 0.707i)12-s + (−0.707 + 0.707i)14-s + 1.00·15-s − 1.00·16-s + (0.707 − 0.707i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4896236624\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4896236624\) |
\(L(1)\) |
|
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 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 - 1.41T + T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + 2iT - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-1 - i)T + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-1 + i)T - iT^{2} \) |
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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
−10.28575046308627946864190954219, −9.588533819693222175183673687480, −8.280410840573720331898018058079, −7.60830839848579287449144212332, −6.92531020987349327793986997360, −6.21183307845880297793203296716, −4.36144758484577729371650016159, −3.74190083982064994031321079220, −2.25844481267753838998040089743, −0.808146649981180763141789924352,
1.34108280495155758038607577854, 3.55255530132479921596922979544, 4.75427825159890925897174862724, 5.35456511869911798075528263888, 6.36791357826312588518274170699, 7.09625253732592273389670699645, 8.474954288370207451266114004223, 8.922597913545683929451104444908, 9.507376332513271597442946892600, 10.56036348914369909976436189427