| L(s) = 1 | − 2-s + 4-s + 3.97·5-s + 3.97·7-s − 8-s − 3.97·10-s − 4.91·11-s + 6.34·13-s − 3.97·14-s + 16-s − 3.91·17-s + 7.78·19-s + 3.97·20-s + 4.91·22-s − 0.326·23-s + 10.8·25-s − 6.34·26-s + 3.97·28-s + 2.16·29-s + 3.23·31-s − 32-s + 3.91·34-s + 15.8·35-s − 3.13·37-s − 7.78·38-s − 3.97·40-s + 2.94·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.77·5-s + 1.50·7-s − 0.353·8-s − 1.25·10-s − 1.48·11-s + 1.76·13-s − 1.06·14-s + 0.250·16-s − 0.949·17-s + 1.78·19-s + 0.889·20-s + 1.04·22-s − 0.0681·23-s + 2.16·25-s − 1.24·26-s + 0.751·28-s + 0.402·29-s + 0.581·31-s − 0.176·32-s + 0.671·34-s + 2.67·35-s − 0.516·37-s − 1.26·38-s − 0.628·40-s + 0.460·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.893498432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.893498432\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 - 3.97T + 5T^{2} \) |
| 7 | \( 1 - 3.97T + 7T^{2} \) |
| 11 | \( 1 + 4.91T + 11T^{2} \) |
| 13 | \( 1 - 6.34T + 13T^{2} \) |
| 17 | \( 1 + 3.91T + 17T^{2} \) |
| 19 | \( 1 - 7.78T + 19T^{2} \) |
| 23 | \( 1 + 0.326T + 23T^{2} \) |
| 29 | \( 1 - 2.16T + 29T^{2} \) |
| 31 | \( 1 - 3.23T + 31T^{2} \) |
| 37 | \( 1 + 3.13T + 37T^{2} \) |
| 41 | \( 1 - 2.94T + 41T^{2} \) |
| 43 | \( 1 + 8.02T + 43T^{2} \) |
| 47 | \( 1 - 0.363T + 47T^{2} \) |
| 53 | \( 1 - 8.28T + 53T^{2} \) |
| 59 | \( 1 + 1.35T + 59T^{2} \) |
| 61 | \( 1 - 2.36T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 16.4T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 3.09T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 7.48T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.004186997739780115695056526715, −7.22509821808303611518724863339, −6.43990652852723958400735857645, −5.61583293987381691348802534232, −5.35849113087514525100326073521, −4.52840339340571898669679666771, −3.15966787147079774013256620211, −2.38446631029095691081337607082, −1.64104143901890803086609887527, −1.04035966786327686969974188151,
1.04035966786327686969974188151, 1.64104143901890803086609887527, 2.38446631029095691081337607082, 3.15966787147079774013256620211, 4.52840339340571898669679666771, 5.35849113087514525100326073521, 5.61583293987381691348802534232, 6.43990652852723958400735857645, 7.22509821808303611518724863339, 8.004186997739780115695056526715
