| L(s) = 1 | − 1.86·2-s − 3·3-s − 4.52·4-s − 2.36·5-s + 5.59·6-s − 4.86·7-s + 23.3·8-s + 9·9-s + 4.40·10-s − 35.1·11-s + 13.5·12-s + 9.07·14-s + 7.08·15-s − 7.35·16-s − 33.1·17-s − 16.7·18-s − 104.·19-s + 10.6·20-s + 14.5·21-s + 65.5·22-s − 86.3·23-s − 70.0·24-s − 119.·25-s − 27·27-s + 22.0·28-s − 118.·29-s − 13.2·30-s + ⋯ |
| L(s) = 1 | − 0.659·2-s − 0.577·3-s − 0.565·4-s − 0.211·5-s + 0.380·6-s − 0.262·7-s + 1.03·8-s + 0.333·9-s + 0.139·10-s − 0.963·11-s + 0.326·12-s + 0.173·14-s + 0.121·15-s − 0.114·16-s − 0.472·17-s − 0.219·18-s − 1.25·19-s + 0.119·20-s + 0.151·21-s + 0.635·22-s − 0.782·23-s − 0.595·24-s − 0.955·25-s − 0.192·27-s + 0.148·28-s − 0.756·29-s − 0.0803·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3723278693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3723278693\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.86T + 8T^{2} \) |
| 5 | \( 1 + 2.36T + 125T^{2} \) |
| 7 | \( 1 + 4.86T + 343T^{2} \) |
| 11 | \( 1 + 35.1T + 1.33e3T^{2} \) |
| 17 | \( 1 + 33.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 86.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 118.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 262.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 59.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 76.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 344.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 415.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 141.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 598.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 791.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 22.1T + 3.00e5T^{2} \) |
| 71 | \( 1 + 599.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 776.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.27e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 493.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.25e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 76.4T + 9.12e5T^{2} \) |
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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
−10.27976598625965183119153408444, −9.834576423937564192429281295940, −8.605909819809659262910921622099, −8.019614239907496078818067172901, −6.95829588901256559140533540666, −5.84450571261277460768712948758, −4.79125945362830518166900124700, −3.87627373910893986795959781263, −2.09547075524925411666245668846, −0.41634657336465688632166560629,
0.41634657336465688632166560629, 2.09547075524925411666245668846, 3.87627373910893986795959781263, 4.79125945362830518166900124700, 5.84450571261277460768712948758, 6.95829588901256559140533540666, 8.019614239907496078818067172901, 8.605909819809659262910921622099, 9.834576423937564192429281295940, 10.27976598625965183119153408444
