L(s) = 1 | − 3.05·3-s + 5-s + 2.86·7-s + 6.31·9-s − 2.92·11-s + 2.59·13-s − 3.05·15-s − 3.00·17-s − 6.44·19-s − 8.73·21-s − 6.65·23-s + 25-s − 10.1·27-s + 0.745·29-s − 6.77·31-s + 8.92·33-s + 2.86·35-s + 3.42·37-s − 7.90·39-s + 10.4·41-s − 9.67·43-s + 6.31·45-s + 1.71·47-s + 1.18·49-s + 9.17·51-s − 1.95·53-s − 2.92·55-s + ⋯ |
L(s) = 1 | − 1.76·3-s + 0.447·5-s + 1.08·7-s + 2.10·9-s − 0.881·11-s + 0.718·13-s − 0.788·15-s − 0.729·17-s − 1.47·19-s − 1.90·21-s − 1.38·23-s + 0.200·25-s − 1.94·27-s + 0.138·29-s − 1.21·31-s + 1.55·33-s + 0.483·35-s + 0.563·37-s − 1.26·39-s + 1.63·41-s − 1.47·43-s + 0.941·45-s + 0.249·47-s + 0.168·49-s + 1.28·51-s − 0.267·53-s − 0.394·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9002714710\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9002714710\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 3.05T + 3T^{2} \) |
| 7 | \( 1 - 2.86T + 7T^{2} \) |
| 11 | \( 1 + 2.92T + 11T^{2} \) |
| 13 | \( 1 - 2.59T + 13T^{2} \) |
| 17 | \( 1 + 3.00T + 17T^{2} \) |
| 19 | \( 1 + 6.44T + 19T^{2} \) |
| 23 | \( 1 + 6.65T + 23T^{2} \) |
| 29 | \( 1 - 0.745T + 29T^{2} \) |
| 31 | \( 1 + 6.77T + 31T^{2} \) |
| 37 | \( 1 - 3.42T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 9.67T + 43T^{2} \) |
| 47 | \( 1 - 1.71T + 47T^{2} \) |
| 53 | \( 1 + 1.95T + 53T^{2} \) |
| 59 | \( 1 - 2.01T + 59T^{2} \) |
| 61 | \( 1 - 4.02T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 1.82T + 73T^{2} \) |
| 79 | \( 1 - 7.86T + 79T^{2} \) |
| 83 | \( 1 + 0.167T + 83T^{2} \) |
| 89 | \( 1 + 3.06T + 89T^{2} \) |
| 97 | \( 1 + 2.34T + 97T^{2} \) |
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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
−7.904026774964655495704677889818, −6.81091206526556125410169582539, −6.39383885521769871022927679800, −5.64498413087394114540770245433, −5.23925295646691588332804957629, −4.45909493670335800464525654132, −3.93499665600209956892693196252, −2.27624362946955968923254478416, −1.69290546059315795829881972161, −0.51613898318832568717081311774,
0.51613898318832568717081311774, 1.69290546059315795829881972161, 2.27624362946955968923254478416, 3.93499665600209956892693196252, 4.45909493670335800464525654132, 5.23925295646691588332804957629, 5.64498413087394114540770245433, 6.39383885521769871022927679800, 6.81091206526556125410169582539, 7.904026774964655495704677889818