L(s) = 1 | − i·5-s − 3.41i·7-s − 2.58·11-s − 3.41·13-s + 1.17i·17-s − 4.82·23-s − 25-s + 6i·29-s − 6.48i·31-s − 3.41·35-s − 9.07·37-s + 11.0i·41-s + 6.82i·43-s + 5.65·47-s − 4.65·49-s + ⋯ |
L(s) = 1 | − 0.447i·5-s − 1.29i·7-s − 0.779·11-s − 0.946·13-s + 0.284i·17-s − 1.00·23-s − 0.200·25-s + 1.11i·29-s − 1.16i·31-s − 0.577·35-s − 1.49·37-s + 1.72i·41-s + 1.04i·43-s + 0.825·47-s − 0.665·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3379721590\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3379721590\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
good | 7 | \( 1 + 3.41iT - 7T^{2} \) |
| 11 | \( 1 + 2.58T + 11T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 - 1.17iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 6.48iT - 31T^{2} \) |
| 37 | \( 1 + 9.07T + 37T^{2} \) |
| 41 | \( 1 - 11.0iT - 41T^{2} \) |
| 43 | \( 1 - 6.82iT - 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 1.17iT - 53T^{2} \) |
| 59 | \( 1 + 6.58T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 14.4iT - 79T^{2} \) |
| 83 | \( 1 + 9.17T + 83T^{2} \) |
| 89 | \( 1 + 4.24iT - 89T^{2} \) |
| 97 | \( 1 - 2.48T + 97T^{2} \) |
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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
−9.183930417254190500444321806524, −8.100586217692038058966235330728, −7.60277155322429224672156892573, −6.79794682135626334160360276781, −5.72482668957035523704622096188, −4.77872358348729169092522851647, −4.09645573458607914689342383210, −2.94443855828417120921757167260, −1.59552567414014799363337751330, −0.12641982256569041312251386921,
2.13403615606214718347597235670, 2.70807452517557273754609515357, 3.94321465814082114241693158567, 5.25294005742020447763546785498, 5.61786075098206753695476386655, 6.77213987707723257656505294708, 7.50986061553617405595987370587, 8.433382451584547715549959545911, 9.074238608412405889362099180842, 10.07059232661073446077601059323