L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 2·7-s + 8-s + 9-s − 3·11-s + 12-s − 6·13-s − 2·14-s + 16-s − 2·17-s + 18-s − 19-s − 2·21-s − 3·22-s − 23-s + 24-s − 6·26-s + 27-s − 2·28-s − 5·29-s + 7·31-s + 32-s − 3·33-s − 2·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.288·12-s − 1.66·13-s − 0.534·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.229·19-s − 0.436·21-s − 0.639·22-s − 0.208·23-s + 0.204·24-s − 1.17·26-s + 0.192·27-s − 0.377·28-s − 0.928·29-s + 1.25·31-s + 0.176·32-s − 0.522·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.217427984817066356930559114708, −7.60622459677158460518702116522, −6.86569603805897577773508143623, −6.15134556862005864870337322514, −5.07777246730808825040461724128, −4.57231071500692709436179613599, −3.44331974409215101364001794232, −2.74533500387991842347800369211, −1.98976766794239613645990104673, 0,
1.98976766794239613645990104673, 2.74533500387991842347800369211, 3.44331974409215101364001794232, 4.57231071500692709436179613599, 5.07777246730808825040461724128, 6.15134556862005864870337322514, 6.86569603805897577773508143623, 7.60622459677158460518702116522, 8.217427984817066356930559114708