| L(s) = 1 | + 2.08·3-s + 2.45·5-s + 0.831·7-s + 1.34·9-s + 4.68·11-s + 5.03·13-s + 5.12·15-s − 1.68·17-s − 2.86·19-s + 1.73·21-s + 4.55·23-s + 1.04·25-s − 3.44·27-s + 5.60·29-s + 9.55·31-s + 9.77·33-s + 2.04·35-s − 10.9·37-s + 10.5·39-s + 3.27·41-s − 4.54·43-s + 3.31·45-s + 5.74·47-s − 6.30·49-s − 3.52·51-s + 6.11·53-s + 11.5·55-s + ⋯ |
| L(s) = 1 | + 1.20·3-s + 1.09·5-s + 0.314·7-s + 0.449·9-s + 1.41·11-s + 1.39·13-s + 1.32·15-s − 0.409·17-s − 0.656·19-s + 0.378·21-s + 0.949·23-s + 0.208·25-s − 0.662·27-s + 1.04·29-s + 1.71·31-s + 1.70·33-s + 0.345·35-s − 1.79·37-s + 1.68·39-s + 0.510·41-s − 0.693·43-s + 0.494·45-s + 0.837·47-s − 0.901·49-s − 0.493·51-s + 0.839·53-s + 1.55·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.080228202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.080228202\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.08T + 3T^{2} \) |
| 5 | \( 1 - 2.45T + 5T^{2} \) |
| 7 | \( 1 - 0.831T + 7T^{2} \) |
| 11 | \( 1 - 4.68T + 11T^{2} \) |
| 13 | \( 1 - 5.03T + 13T^{2} \) |
| 17 | \( 1 + 1.68T + 17T^{2} \) |
| 19 | \( 1 + 2.86T + 19T^{2} \) |
| 23 | \( 1 - 4.55T + 23T^{2} \) |
| 29 | \( 1 - 5.60T + 29T^{2} \) |
| 31 | \( 1 - 9.55T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 3.27T + 41T^{2} \) |
| 43 | \( 1 + 4.54T + 43T^{2} \) |
| 47 | \( 1 - 5.74T + 47T^{2} \) |
| 53 | \( 1 - 6.11T + 53T^{2} \) |
| 59 | \( 1 + 2.50T + 59T^{2} \) |
| 61 | \( 1 + 7.60T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 2.76T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 - 1.87T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−8.140659119158302367006236957654, −7.05935063101620399106233006824, −6.38267074747159054048302824110, −6.02187697446463752323885613395, −4.92402052335020653043559102880, −4.14421588901602146968236551584, −3.40805185267068987103129239214, −2.66264938810380976992470001574, −1.77015098464444986892145290501, −1.19595133970979089307351655046,
1.19595133970979089307351655046, 1.77015098464444986892145290501, 2.66264938810380976992470001574, 3.40805185267068987103129239214, 4.14421588901602146968236551584, 4.92402052335020653043559102880, 6.02187697446463752323885613395, 6.38267074747159054048302824110, 7.05935063101620399106233006824, 8.140659119158302367006236957654
