| L(s) = 1 | + 5·3-s − 19·5-s + 3·7-s − 2·9-s − 2·11-s + 13·13-s − 95·15-s + 77·17-s − 58·19-s + 15·21-s − 76·23-s + 236·25-s − 145·27-s + 6·29-s + 292·31-s − 10·33-s − 57·35-s − 207·37-s + 65·39-s + 240·41-s − 317·43-s + 38·45-s + 375·47-s − 334·49-s + 385·51-s + 692·53-s + 38·55-s + ⋯ |
| L(s) = 1 | + 0.962·3-s − 1.69·5-s + 0.161·7-s − 0.0740·9-s − 0.0548·11-s + 0.277·13-s − 1.63·15-s + 1.09·17-s − 0.700·19-s + 0.155·21-s − 0.689·23-s + 1.88·25-s − 1.03·27-s + 0.0384·29-s + 1.69·31-s − 0.0527·33-s − 0.275·35-s − 0.919·37-s + 0.266·39-s + 0.914·41-s − 1.12·43-s + 0.125·45-s + 1.16·47-s − 0.973·49-s + 1.05·51-s + 1.79·53-s + 0.0931·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.825968000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.825968000\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - p T \) |
| good | 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 5 | \( 1 + 19 T + p^{3} T^{2} \) |
| 7 | \( 1 - 3 T + p^{3} T^{2} \) |
| 11 | \( 1 + 2 T + p^{3} T^{2} \) |
| 17 | \( 1 - 77 T + p^{3} T^{2} \) |
| 19 | \( 1 + 58 T + p^{3} T^{2} \) |
| 23 | \( 1 + 76 T + p^{3} T^{2} \) |
| 29 | \( 1 - 6 T + p^{3} T^{2} \) |
| 31 | \( 1 - 292 T + p^{3} T^{2} \) |
| 37 | \( 1 + 207 T + p^{3} T^{2} \) |
| 41 | \( 1 - 240 T + p^{3} T^{2} \) |
| 43 | \( 1 + 317 T + p^{3} T^{2} \) |
| 47 | \( 1 - 375 T + p^{3} T^{2} \) |
| 53 | \( 1 - 692 T + p^{3} T^{2} \) |
| 59 | \( 1 - 214 T + p^{3} T^{2} \) |
| 61 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 67 | \( 1 - 782 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1057 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1174 T + p^{3} T^{2} \) |
| 79 | \( 1 + 892 T + p^{3} T^{2} \) |
| 83 | \( 1 - 704 T + p^{3} T^{2} \) |
| 89 | \( 1 - 6 T + p^{3} T^{2} \) |
| 97 | \( 1 - 830 T + p^{3} T^{2} \) |
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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
−9.751337333900448937411325019774, −8.547826507927824777458276813255, −8.243850230091312876490731517223, −7.60268565652320666338760423282, −6.58187762536616659676182599703, −5.22707662283289051760328318190, −4.02110140195504520010367472335, −3.51282270613976165425716670769, −2.41326882379388916588095305629, −0.70546216817578360793524888031,
0.70546216817578360793524888031, 2.41326882379388916588095305629, 3.51282270613976165425716670769, 4.02110140195504520010367472335, 5.22707662283289051760328318190, 6.58187762536616659676182599703, 7.60268565652320666338760423282, 8.243850230091312876490731517223, 8.547826507927824777458276813255, 9.751337333900448937411325019774
