L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 2·7-s − 8-s + 9-s + 10-s + 0.464·11-s − 12-s + 2·14-s + 15-s + 16-s + 4·17-s − 18-s − 0.535·19-s − 20-s + 2·21-s − 0.464·22-s − 0.267·23-s + 24-s + 25-s − 27-s − 2·28-s + 3.73·29-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.139·11-s − 0.288·12-s + 0.534·14-s + 0.258·15-s + 0.250·16-s + 0.970·17-s − 0.235·18-s − 0.122·19-s − 0.223·20-s + 0.436·21-s − 0.0989·22-s − 0.0558·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s − 0.377·28-s + 0.693·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7128015949\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7128015949\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 0.464T + 11T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 0.535T + 19T^{2} \) |
| 23 | \( 1 + 0.267T + 23T^{2} \) |
| 29 | \( 1 - 3.73T + 29T^{2} \) |
| 31 | \( 1 + 1.73T + 31T^{2} \) |
| 37 | \( 1 - 1.19T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 1.92T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 1.53T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 4.53T + 67T^{2} \) |
| 71 | \( 1 + 8.39T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 0.0717T + 79T^{2} \) |
| 83 | \( 1 - 4.92T + 83T^{2} \) |
| 89 | \( 1 - 7.46T + 89T^{2} \) |
| 97 | \( 1 + 7.46T + 97T^{2} \) |
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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
−8.121286495555366771519048716328, −7.61366979273131513139950851041, −6.74490023567640168181283829848, −6.29262852329280302628441366762, −5.46148624164077495930780017736, −4.57938699461460910879882299454, −3.58256311170217966191857119843, −2.89745681348894234828070251987, −1.60911771537227010463981368190, −0.53771870815811906812815603358,
0.53771870815811906812815603358, 1.60911771537227010463981368190, 2.89745681348894234828070251987, 3.58256311170217966191857119843, 4.57938699461460910879882299454, 5.46148624164077495930780017736, 6.29262852329280302628441366762, 6.74490023567640168181283829848, 7.61366979273131513139950851041, 8.121286495555366771519048716328