| L(s) = 1 | − 1.31·3-s − 1.28·5-s + 1.02·7-s − 1.26·9-s − 3.40·11-s − 6.32·13-s + 1.69·15-s + 5.36·17-s + 2.29·19-s − 1.34·21-s − 6.25·23-s − 3.35·25-s + 5.61·27-s − 8.07·29-s − 1.34·31-s + 4.48·33-s − 1.31·35-s − 7.74·37-s + 8.32·39-s + 9.95·41-s + 1.55·43-s + 1.62·45-s + 11.5·47-s − 5.95·49-s − 7.06·51-s − 8.43·53-s + 4.37·55-s + ⋯ |
| L(s) = 1 | − 0.760·3-s − 0.574·5-s + 0.385·7-s − 0.421·9-s − 1.02·11-s − 1.75·13-s + 0.436·15-s + 1.30·17-s + 0.525·19-s − 0.293·21-s − 1.30·23-s − 0.670·25-s + 1.08·27-s − 1.49·29-s − 0.240·31-s + 0.781·33-s − 0.221·35-s − 1.27·37-s + 1.33·39-s + 1.55·41-s + 0.236·43-s + 0.242·45-s + 1.68·47-s − 0.851·49-s − 0.988·51-s − 1.15·53-s + 0.590·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5215834147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5215834147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 1.31T + 3T^{2} \) |
| 5 | \( 1 + 1.28T + 5T^{2} \) |
| 7 | \( 1 - 1.02T + 7T^{2} \) |
| 11 | \( 1 + 3.40T + 11T^{2} \) |
| 13 | \( 1 + 6.32T + 13T^{2} \) |
| 17 | \( 1 - 5.36T + 17T^{2} \) |
| 19 | \( 1 - 2.29T + 19T^{2} \) |
| 23 | \( 1 + 6.25T + 23T^{2} \) |
| 29 | \( 1 + 8.07T + 29T^{2} \) |
| 31 | \( 1 + 1.34T + 31T^{2} \) |
| 37 | \( 1 + 7.74T + 37T^{2} \) |
| 41 | \( 1 - 9.95T + 41T^{2} \) |
| 43 | \( 1 - 1.55T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 8.43T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 0.212T + 61T^{2} \) |
| 67 | \( 1 + 1.62T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 1.94T + 83T^{2} \) |
| 89 | \( 1 + 4.10T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−8.180830576019184733431387983285, −7.55823932304405376833985724432, −7.33505000073012128729417000054, −5.92084480023219059782941657074, −5.49683277458708958134031944551, −4.89161675897817443268350098247, −3.93013195606143784075159772713, −2.93229139699040334576316888504, −2.00130107261522118106772228459, −0.40647327851053135596870135393,
0.40647327851053135596870135393, 2.00130107261522118106772228459, 2.93229139699040334576316888504, 3.93013195606143784075159772713, 4.89161675897817443268350098247, 5.49683277458708958134031944551, 5.92084480023219059782941657074, 7.33505000073012128729417000054, 7.55823932304405376833985724432, 8.180830576019184733431387983285
