L(s) = 1 | − 2-s + 4-s − 5-s + 2.92·7-s − 8-s + 10-s + 0.926·11-s + 6.21·13-s − 2.92·14-s + 16-s − 7.09·17-s − 1.28·19-s − 20-s − 0.926·22-s + 3.80·23-s + 25-s − 6.21·26-s + 2.92·28-s + 8.42·29-s + 31-s − 32-s + 7.09·34-s − 2.92·35-s − 7.97·37-s + 1.28·38-s + 40-s + 8.42·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.10·7-s − 0.353·8-s + 0.316·10-s + 0.279·11-s + 1.72·13-s − 0.782·14-s + 0.250·16-s − 1.72·17-s − 0.295·19-s − 0.223·20-s − 0.197·22-s + 0.793·23-s + 0.200·25-s − 1.21·26-s + 0.553·28-s + 1.56·29-s + 0.179·31-s − 0.176·32-s + 1.21·34-s − 0.494·35-s − 1.31·37-s + 0.208·38-s + 0.158·40-s + 1.31·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.490517031\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.490517031\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 - 2.92T + 7T^{2} \) |
| 11 | \( 1 - 0.926T + 11T^{2} \) |
| 13 | \( 1 - 6.21T + 13T^{2} \) |
| 17 | \( 1 + 7.09T + 17T^{2} \) |
| 19 | \( 1 + 1.28T + 19T^{2} \) |
| 23 | \( 1 - 3.80T + 23T^{2} \) |
| 29 | \( 1 - 8.42T + 29T^{2} \) |
| 37 | \( 1 + 7.97T + 37T^{2} \) |
| 41 | \( 1 - 8.42T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 5.75T + 47T^{2} \) |
| 53 | \( 1 + 5.14T + 53T^{2} \) |
| 59 | \( 1 + 8.11T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 - 5.39T + 67T^{2} \) |
| 71 | \( 1 + 7.50T + 71T^{2} \) |
| 73 | \( 1 - 3.43T + 73T^{2} \) |
| 79 | \( 1 + 0.472T + 79T^{2} \) |
| 83 | \( 1 + 7.61T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 8.57T + 97T^{2} \) |
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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
−8.865047974242424666159273698826, −8.201947170833163426565528369465, −7.46382536087789089556534335883, −6.58555779480055710031235623180, −5.99106178563630460917042034188, −4.72444519509927448396174134588, −4.15925891827925806725940895572, −2.98607088277888542708005470902, −1.83987922280783240194920090603, −0.888767666197895784532378805436,
0.888767666197895784532378805436, 1.83987922280783240194920090603, 2.98607088277888542708005470902, 4.15925891827925806725940895572, 4.72444519509927448396174134588, 5.99106178563630460917042034188, 6.58555779480055710031235623180, 7.46382536087789089556534335883, 8.201947170833163426565528369465, 8.865047974242424666159273698826