L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 4·11-s − 12-s + 2·13-s + 14-s − 15-s + 16-s + 2·17-s − 18-s + 20-s + 21-s + 4·22-s − 8·23-s + 24-s + 25-s − 2·26-s − 27-s − 28-s + 2·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s + 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.223·20-s + 0.218·21-s + 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.09603199550571, −13.93289392978521, −13.13103599019601, −12.82698948998854, −12.19708413136307, −11.85021666156573, −11.12449601069171, −10.70441193117749, −10.27367386319319, −9.759446557733586, −9.552875516076853, −8.510129750047726, −8.343805393957350, −7.723675485373249, −7.065718686433825, −6.634565818629241, −5.949105732390926, −5.584112720393591, −5.134152047005765, −4.209582094089969, −3.652907463873193, −2.843274427969113, −2.268462418133431, −1.592446381738218, −0.7533162933479332, 0,
0.7533162933479332, 1.592446381738218, 2.268462418133431, 2.843274427969113, 3.652907463873193, 4.209582094089969, 5.134152047005765, 5.584112720393591, 5.949105732390926, 6.634565818629241, 7.065718686433825, 7.723675485373249, 8.343805393957350, 8.510129750047726, 9.552875516076853, 9.759446557733586, 10.27367386319319, 10.70441193117749, 11.12449601069171, 11.85021666156573, 12.19708413136307, 12.82698948998854, 13.13103599019601, 13.93289392978521, 14.09603199550571