L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s − 2·7-s + 3·8-s + 9-s + 10-s − 2·11-s − 12-s + 4·13-s + 2·14-s − 15-s − 16-s + 2·17-s − 18-s + 20-s − 2·21-s + 2·22-s − 4·23-s + 3·24-s + 25-s − 4·26-s + 27-s + 2·28-s − 4·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s + 1.10·13-s + 0.534·14-s − 0.258·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.223·20-s − 0.436·21-s + 0.426·22-s − 0.834·23-s + 0.612·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s + 0.377·28-s − 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.933571531297479789633183372924, −7.44287345072152491475459538796, −6.54229826136665453444957630969, −5.68165521799718833065526766039, −4.81235677539192812176498278300, −3.79376989874905699484027875919, −3.49719173564619629745176426371, −2.27642403463647267927236878892, −1.14662848613992672908983132209, 0,
1.14662848613992672908983132209, 2.27642403463647267927236878892, 3.49719173564619629745176426371, 3.79376989874905699484027875919, 4.81235677539192812176498278300, 5.68165521799718833065526766039, 6.54229826136665453444957630969, 7.44287345072152491475459538796, 7.933571531297479789633183372924