L(s) = 1 | + 2-s + 4-s + 4.24·7-s + 8-s + 1.42·11-s − 6.91·13-s + 4.24·14-s + 16-s − 5.10·17-s + 19-s + 1.42·22-s + 3.67·23-s − 6.91·26-s + 4.24·28-s + 8.10·29-s − 1.28·31-s + 32-s − 5.10·34-s − 0.856·37-s + 38-s − 8.01·41-s + 3.57·43-s + 1.42·44-s + 3.67·46-s − 3.81·47-s + 11.0·49-s − 6.91·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.60·7-s + 0.353·8-s + 0.430·11-s − 1.91·13-s + 1.13·14-s + 0.250·16-s − 1.23·17-s + 0.229·19-s + 0.304·22-s + 0.765·23-s − 1.35·26-s + 0.802·28-s + 1.50·29-s − 0.231·31-s + 0.176·32-s − 0.874·34-s − 0.140·37-s + 0.162·38-s − 1.25·41-s + 0.544·43-s + 0.215·44-s + 0.541·46-s − 0.556·47-s + 1.57·49-s − 0.959·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.947562825\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.947562825\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 - 1.42T + 11T^{2} \) |
| 13 | \( 1 + 6.91T + 13T^{2} \) |
| 17 | \( 1 + 5.10T + 17T^{2} \) |
| 23 | \( 1 - 3.67T + 23T^{2} \) |
| 29 | \( 1 - 8.10T + 29T^{2} \) |
| 31 | \( 1 + 1.28T + 31T^{2} \) |
| 37 | \( 1 + 0.856T + 37T^{2} \) |
| 41 | \( 1 + 8.01T + 41T^{2} \) |
| 43 | \( 1 - 3.57T + 43T^{2} \) |
| 47 | \( 1 + 3.81T + 47T^{2} \) |
| 53 | \( 1 - 9.06T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 8.20T + 61T^{2} \) |
| 67 | \( 1 - 4.38T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 5.38T + 73T^{2} \) |
| 79 | \( 1 - 2.14T + 79T^{2} \) |
| 83 | \( 1 - 1.04T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 - 6.81T + 97T^{2} \) |
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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
−7.65008002245673776211333874104, −7.00364219325079098613575780937, −6.54962312579035980686255608155, −5.32018264938890129114970788424, −4.95337014137384165271349781782, −4.52480889681891354924898888846, −3.62539071905029847227501644835, −2.42900083682186409095468063648, −2.10130147266389542894321434480, −0.880073733301537637760853003870,
0.880073733301537637760853003870, 2.10130147266389542894321434480, 2.42900083682186409095468063648, 3.62539071905029847227501644835, 4.52480889681891354924898888846, 4.95337014137384165271349781782, 5.32018264938890129114970788424, 6.54962312579035980686255608155, 7.00364219325079098613575780937, 7.65008002245673776211333874104