L(s) = 1 | − 1.76·2-s + 0.777·3-s + 1.10·4-s − 1.36·6-s − 0.840·7-s + 1.58·8-s − 2.39·9-s − 3.84·11-s + 0.856·12-s + 6.84·13-s + 1.48·14-s − 4.98·16-s − 4.50·17-s + 4.21·18-s − 4.75·19-s − 0.653·21-s + 6.78·22-s + 1.21·23-s + 1.23·24-s − 12.0·26-s − 4.19·27-s − 0.926·28-s − 1.98·29-s + 3.16·31-s + 5.62·32-s − 2.99·33-s + 7.93·34-s + ⋯ |
L(s) = 1 | − 1.24·2-s + 0.448·3-s + 0.550·4-s − 0.559·6-s − 0.317·7-s + 0.559·8-s − 0.798·9-s − 1.16·11-s + 0.247·12-s + 1.89·13-s + 0.395·14-s − 1.24·16-s − 1.09·17-s + 0.994·18-s − 1.09·19-s − 0.142·21-s + 1.44·22-s + 0.253·23-s + 0.251·24-s − 2.36·26-s − 0.807·27-s − 0.175·28-s − 0.368·29-s + 0.568·31-s + 0.994·32-s − 0.521·33-s + 1.36·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6030269242\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6030269242\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 1.76T + 2T^{2} \) |
| 3 | \( 1 - 0.777T + 3T^{2} \) |
| 7 | \( 1 + 0.840T + 7T^{2} \) |
| 11 | \( 1 + 3.84T + 11T^{2} \) |
| 13 | \( 1 - 6.84T + 13T^{2} \) |
| 17 | \( 1 + 4.50T + 17T^{2} \) |
| 19 | \( 1 + 4.75T + 19T^{2} \) |
| 23 | \( 1 - 1.21T + 23T^{2} \) |
| 29 | \( 1 + 1.98T + 29T^{2} \) |
| 31 | \( 1 - 3.16T + 31T^{2} \) |
| 37 | \( 1 + 4.75T + 37T^{2} \) |
| 41 | \( 1 + 0.0513T + 41T^{2} \) |
| 43 | \( 1 + 4.71T + 43T^{2} \) |
| 47 | \( 1 - 2.00T + 47T^{2} \) |
| 53 | \( 1 - 8.94T + 53T^{2} \) |
| 59 | \( 1 - 7.53T + 59T^{2} \) |
| 61 | \( 1 - 2.30T + 61T^{2} \) |
| 67 | \( 1 - 3.71T + 67T^{2} \) |
| 71 | \( 1 + 5.82T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 5.99T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 8.96T + 89T^{2} \) |
| 97 | \( 1 - 1.15T + 97T^{2} \) |
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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
−8.395292711778025616716887168410, −7.69944112506999347604008397173, −6.79348442695868700017354359101, −6.18109847633187297271601302035, −5.31582150826209234794552895848, −4.32587625786687699538733948599, −3.48871969629317116687494045234, −2.53671291047589837134664633642, −1.76391569211927330198134229541, −0.47073302569378394007493285080,
0.47073302569378394007493285080, 1.76391569211927330198134229541, 2.53671291047589837134664633642, 3.48871969629317116687494045234, 4.32587625786687699538733948599, 5.31582150826209234794552895848, 6.18109847633187297271601302035, 6.79348442695868700017354359101, 7.69944112506999347604008397173, 8.395292711778025616716887168410