L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s − 2·9-s − 3·11-s − 12-s − 2·13-s + 14-s + 16-s − 3·17-s − 2·18-s − 8·19-s − 21-s − 3·22-s − 24-s − 2·26-s + 5·27-s + 28-s − 6·29-s − 10·31-s + 32-s + 3·33-s − 3·34-s − 2·36-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.904·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 1.83·19-s − 0.218·21-s − 0.639·22-s − 0.204·24-s − 0.392·26-s + 0.962·27-s + 0.188·28-s − 1.11·29-s − 1.79·31-s + 0.176·32-s + 0.522·33-s − 0.514·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−13.59147765027246, −12.96844736263331, −12.69606552217918, −12.33821886215517, −11.74675504710551, −11.05481036454502, −10.96169037120842, −10.67064696587355, −9.995874635382940, −9.284899578780880, −8.758279710307617, −8.434117197991167, −7.693525845422992, −7.198029506436467, −6.952014249195883, −5.934358224913928, −5.828455484052066, −5.461884981317259, −4.717771773273665, −4.246851514510400, −3.937711870043750, −2.942893154250354, −2.436446225153520, −2.138957148327456, −1.244728484095083, 0, 0,
1.244728484095083, 2.138957148327456, 2.436446225153520, 2.942893154250354, 3.937711870043750, 4.246851514510400, 4.717771773273665, 5.461884981317259, 5.828455484052066, 5.934358224913928, 6.952014249195883, 7.198029506436467, 7.693525845422992, 8.434117197991167, 8.758279710307617, 9.284899578780880, 9.995874635382940, 10.67064696587355, 10.96169037120842, 11.05481036454502, 11.74675504710551, 12.33821886215517, 12.69606552217918, 12.96844736263331, 13.59147765027246