L(s) = 1 | + 2.03·3-s + 4.46·5-s + 0.828·7-s + 1.13·9-s + 0.0966·11-s + 1.53·13-s + 9.07·15-s − 3.59·17-s + 8.33·19-s + 1.68·21-s + 2.93·23-s + 14.9·25-s − 3.79·27-s + 1.38·29-s − 2.48·31-s + 0.196·33-s + 3.69·35-s − 3.06·37-s + 3.12·39-s − 5.09·41-s − 12.0·43-s + 5.05·45-s − 3.75·47-s − 6.31·49-s − 7.30·51-s + 5.98·53-s + 0.431·55-s + ⋯ |
L(s) = 1 | + 1.17·3-s + 1.99·5-s + 0.313·7-s + 0.378·9-s + 0.0291·11-s + 0.426·13-s + 2.34·15-s − 0.871·17-s + 1.91·19-s + 0.367·21-s + 0.611·23-s + 2.98·25-s − 0.730·27-s + 0.256·29-s − 0.446·31-s + 0.0342·33-s + 0.625·35-s − 0.504·37-s + 0.501·39-s − 0.794·41-s − 1.84·43-s + 0.754·45-s − 0.547·47-s − 0.901·49-s − 1.02·51-s + 0.821·53-s + 0.0581·55-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
\(L(1)\) |
\(\approx\) |
\(4.647709072\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.647709072\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.03T + 3T^{2} \) |
| 5 | \( 1 - 4.46T + 5T^{2} \) |
| 7 | \( 1 - 0.828T + 7T^{2} \) |
| 11 | \( 1 - 0.0966T + 11T^{2} \) |
| 13 | \( 1 - 1.53T + 13T^{2} \) |
| 17 | \( 1 + 3.59T + 17T^{2} \) |
| 19 | \( 1 - 8.33T + 19T^{2} \) |
| 23 | \( 1 - 2.93T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 + 2.48T + 31T^{2} \) |
| 37 | \( 1 + 3.06T + 37T^{2} \) |
| 41 | \( 1 + 5.09T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 3.75T + 47T^{2} \) |
| 53 | \( 1 - 5.98T + 53T^{2} \) |
| 59 | \( 1 - 1.55T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 1.19T + 67T^{2} \) |
| 71 | \( 1 + 5.48T + 71T^{2} \) |
| 73 | \( 1 - 5.93T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 + 7.41T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−8.695513591330745483807597509280, −7.85561908890381062990494791237, −6.90991341563468220833483682310, −6.31567046547131633354882786576, −5.33064407510800716926984271634, −4.95120284214288698801550515334, −3.47409875320108755735511587672, −2.88102092686842330680405565995, −1.98601260845425265713941480362, −1.36108621116723172056489891470,
1.36108621116723172056489891470, 1.98601260845425265713941480362, 2.88102092686842330680405565995, 3.47409875320108755735511587672, 4.95120284214288698801550515334, 5.33064407510800716926984271634, 6.31567046547131633354882786576, 6.90991341563468220833483682310, 7.85561908890381062990494791237, 8.695513591330745483807597509280