| L(s) = 1 | − 0.950·3-s + 3.01·5-s + 0.279·7-s − 2.09·9-s + 3.32·11-s + 2.71·13-s − 2.86·15-s + 6.56·17-s + 6.07·19-s − 0.265·21-s + 6.35·23-s + 4.09·25-s + 4.84·27-s − 4.96·29-s − 8.98·31-s − 3.16·33-s + 0.842·35-s − 5.88·37-s − 2.57·39-s − 5.52·41-s + 12.0·43-s − 6.32·45-s + 3.77·47-s − 6.92·49-s − 6.24·51-s − 12.2·53-s + 10.0·55-s + ⋯ |
| L(s) = 1 | − 0.548·3-s + 1.34·5-s + 0.105·7-s − 0.698·9-s + 1.00·11-s + 0.752·13-s − 0.740·15-s + 1.59·17-s + 1.39·19-s − 0.0579·21-s + 1.32·23-s + 0.818·25-s + 0.932·27-s − 0.921·29-s − 1.61·31-s − 0.550·33-s + 0.142·35-s − 0.967·37-s − 0.413·39-s − 0.863·41-s + 1.84·43-s − 0.942·45-s + 0.550·47-s − 0.988·49-s − 0.873·51-s − 1.68·53-s + 1.35·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.438097404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.438097404\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 0.950T + 3T^{2} \) |
| 5 | \( 1 - 3.01T + 5T^{2} \) |
| 7 | \( 1 - 0.279T + 7T^{2} \) |
| 11 | \( 1 - 3.32T + 11T^{2} \) |
| 13 | \( 1 - 2.71T + 13T^{2} \) |
| 17 | \( 1 - 6.56T + 17T^{2} \) |
| 19 | \( 1 - 6.07T + 19T^{2} \) |
| 23 | \( 1 - 6.35T + 23T^{2} \) |
| 29 | \( 1 + 4.96T + 29T^{2} \) |
| 31 | \( 1 + 8.98T + 31T^{2} \) |
| 37 | \( 1 + 5.88T + 37T^{2} \) |
| 41 | \( 1 + 5.52T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 - 3.77T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 5.60T + 59T^{2} \) |
| 61 | \( 1 - 6.29T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 0.810T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 2.69T + 83T^{2} \) |
| 89 | \( 1 - 3.51T + 89T^{2} \) |
| 97 | \( 1 - 0.120T + 97T^{2} \) |
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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
−8.672078164965324312600363101132, −7.57392127697859443213319203523, −6.86994142128733099634848564321, −5.99185666776335012640342348875, −5.54985093328125582710681516969, −5.11089306914058371304168810242, −3.66190903409942886339597163460, −3.04854789395061157834231513166, −1.71636701083513043185585907247, −1.01693176912945731645545914052,
1.01693176912945731645545914052, 1.71636701083513043185585907247, 3.04854789395061157834231513166, 3.66190903409942886339597163460, 5.11089306914058371304168810242, 5.54985093328125582710681516969, 5.99185666776335012640342348875, 6.86994142128733099634848564321, 7.57392127697859443213319203523, 8.672078164965324312600363101132
