L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s + 2·11-s + 12-s − 13-s + 14-s + 16-s + 3·17-s + 18-s + 21-s + 2·22-s − 23-s + 24-s − 26-s + 27-s + 28-s − 5·29-s + 7·31-s + 32-s + 2·33-s + 3·34-s + 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 + 1/3·9-s + 0.603·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 0.218·21-s + 0.426·22-s − 0.208·23-s + 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.188·28-s − 0.928·29-s + 1.25·31-s + 0.176·32-s + 0.348·33-s + 0.514·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.281460088\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.281460088\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 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
−9.917916465692438028996286745846, −9.095393434458288487667384425418, −8.123824488175672723866053992296, −7.42947157027638079601857823085, −6.49956657169083240946293873136, −5.53940659626655742515496994020, −4.55212546947813382789009153537, −3.70207964125960752967601340357, −2.68218547395085973452320822196, −1.46948062712831077823931321294,
1.46948062712831077823931321294, 2.68218547395085973452320822196, 3.70207964125960752967601340357, 4.55212546947813382789009153537, 5.53940659626655742515496994020, 6.49956657169083240946293873136, 7.42947157027638079601857823085, 8.123824488175672723866053992296, 9.095393434458288487667384425418, 9.917916465692438028996286745846