L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 4·11-s − 2·13-s + 14-s + 16-s + 6·17-s − 20-s + 4·22-s − 23-s + 25-s − 2·26-s + 28-s + 2·29-s + 8·31-s + 32-s + 6·34-s − 35-s + 2·37-s − 40-s − 2·41-s − 4·43-s + 4·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s + 1.20·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.223·20-s + 0.852·22-s − 0.208·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s + 0.371·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s − 0.169·35-s + 0.328·37-s − 0.158·40-s − 0.312·41-s − 0.609·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.071771041\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.071771041\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.95423452958340, −15.48021481859309, −14.87556521408631, −14.32564382880623, −14.09936555214170, −13.40974586948279, −12.48806557296377, −12.25768692817969, −11.62430810881076, −11.37405942313323, −10.30010034147894, −10.06654607118176, −9.177335000286723, −8.562878797517473, −7.698685967325302, −7.512100154733404, −6.482486635992958, −6.182021443460199, −5.221784718204300, −4.703769651409949, −4.019785923503346, −3.381251356506568, −2.666145308445428, −1.625202893892977, −0.8444662016852640,
0.8444662016852640, 1.625202893892977, 2.666145308445428, 3.381251356506568, 4.019785923503346, 4.703769651409949, 5.221784718204300, 6.182021443460199, 6.482486635992958, 7.512100154733404, 7.698685967325302, 8.562878797517473, 9.177335000286723, 10.06654607118176, 10.30010034147894, 11.37405942313323, 11.62430810881076, 12.25768692817969, 12.48806557296377, 13.40974586948279, 14.09936555214170, 14.32564382880623, 14.87556521408631, 15.48021481859309, 15.95423452958340