L(s) = 1 | − 1.29·2-s − 0.319·4-s − 2.25·7-s + 3.00·8-s − 4.86·11-s − 13-s + 2.92·14-s − 3.25·16-s − 6.57·17-s − 7.87·19-s + 6.30·22-s − 1.04·23-s + 1.29·26-s + 0.721·28-s + 0.639·31-s − 1.78·32-s + 8.51·34-s + 8.89·37-s + 10.2·38-s − 0.818·41-s + 11.8·43-s + 1.55·44-s + 1.36·46-s + 5.90·47-s − 1.89·49-s + 0.319·52-s − 9.00·53-s + ⋯ |
L(s) = 1 | − 0.916·2-s − 0.159·4-s − 0.853·7-s + 1.06·8-s − 1.46·11-s − 0.277·13-s + 0.782·14-s − 0.814·16-s − 1.59·17-s − 1.80·19-s + 1.34·22-s − 0.218·23-s + 0.254·26-s + 0.136·28-s + 0.114·31-s − 0.316·32-s + 1.46·34-s + 1.46·37-s + 1.65·38-s − 0.127·41-s + 1.81·43-s + 0.234·44-s + 0.200·46-s + 0.862·47-s − 0.271·49-s + 0.0443·52-s − 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2730103015\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2730103015\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + 1.29T + 2T^{2} \) |
| 7 | \( 1 + 2.25T + 7T^{2} \) |
| 11 | \( 1 + 4.86T + 11T^{2} \) |
| 17 | \( 1 + 6.57T + 17T^{2} \) |
| 19 | \( 1 + 7.87T + 19T^{2} \) |
| 23 | \( 1 + 1.04T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 0.639T + 31T^{2} \) |
| 37 | \( 1 - 8.89T + 37T^{2} \) |
| 41 | \( 1 + 0.818T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 - 5.90T + 47T^{2} \) |
| 53 | \( 1 + 9.00T + 53T^{2} \) |
| 59 | \( 1 + 7.23T + 59T^{2} \) |
| 61 | \( 1 + 6.89T + 61T^{2} \) |
| 67 | \( 1 - 7.36T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 9.75T + 73T^{2} \) |
| 79 | \( 1 - 9.49T + 79T^{2} \) |
| 83 | \( 1 - 0.724T + 83T^{2} \) |
| 89 | \( 1 + 7.78T + 89T^{2} \) |
| 97 | \( 1 - 4.38T + 97T^{2} \) |
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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
−8.893821080064825346006731221593, −8.062334425913279737342123747266, −7.51671774441619435550398784299, −6.57698330010654833114957852832, −5.89098294088463657656756449414, −4.65798923454218205519322180146, −4.24264862859820940198942012946, −2.79625761663252193615287078265, −2.05599407474302660404907442794, −0.34647198057210361916800626580,
0.34647198057210361916800626580, 2.05599407474302660404907442794, 2.79625761663252193615287078265, 4.24264862859820940198942012946, 4.65798923454218205519322180146, 5.89098294088463657656756449414, 6.57698330010654833114957852832, 7.51671774441619435550398784299, 8.062334425913279737342123747266, 8.893821080064825346006731221593