| L(s) = 1 | + 8·3-s + 5·5-s + 4·7-s + 37·9-s − 12·11-s − 58·13-s + 40·15-s + 66·17-s + 100·19-s + 32·21-s − 132·23-s + 25·25-s + 80·27-s − 90·29-s − 152·31-s − 96·33-s + 20·35-s − 34·37-s − 464·39-s − 438·41-s − 32·43-s + 185·45-s + 204·47-s − 327·49-s + 528·51-s + 222·53-s − 60·55-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 0.447·5-s + 0.215·7-s + 1.37·9-s − 0.328·11-s − 1.23·13-s + 0.688·15-s + 0.941·17-s + 1.20·19-s + 0.332·21-s − 1.19·23-s + 1/5·25-s + 0.570·27-s − 0.576·29-s − 0.880·31-s − 0.506·33-s + 0.0965·35-s − 0.151·37-s − 1.90·39-s − 1.66·41-s − 0.113·43-s + 0.612·45-s + 0.633·47-s − 0.953·49-s + 1.44·51-s + 0.575·53-s − 0.147·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.429699699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.429699699\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 100 T + p^{3} T^{2} \) |
| 23 | \( 1 + 132 T + p^{3} T^{2} \) |
| 29 | \( 1 + 90 T + p^{3} T^{2} \) |
| 31 | \( 1 + 152 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 438 T + p^{3} T^{2} \) |
| 43 | \( 1 + 32 T + p^{3} T^{2} \) |
| 47 | \( 1 - 204 T + p^{3} T^{2} \) |
| 53 | \( 1 - 222 T + p^{3} T^{2} \) |
| 59 | \( 1 + 420 T + p^{3} T^{2} \) |
| 61 | \( 1 - 902 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1024 T + p^{3} T^{2} \) |
| 71 | \( 1 + 432 T + p^{3} T^{2} \) |
| 73 | \( 1 - 362 T + p^{3} T^{2} \) |
| 79 | \( 1 - 160 T + p^{3} T^{2} \) |
| 83 | \( 1 + 72 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1106 T + p^{3} T^{2} \) |
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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
−14.12861087180646197698138208358, −13.09824696524040362830843831189, −11.91572014390373431800958917143, −10.09940456155570379724150282023, −9.449872830658836016599231284185, −8.151066893381357743130569617078, −7.30316245764523480708035426920, −5.28509666936803952638369361029, −3.45189849298215772808839954050, −2.07950011495651059749197235959,
2.07950011495651059749197235959, 3.45189849298215772808839954050, 5.28509666936803952638369361029, 7.30316245764523480708035426920, 8.151066893381357743130569617078, 9.449872830658836016599231284185, 10.09940456155570379724150282023, 11.91572014390373431800958917143, 13.09824696524040362830843831189, 14.12861087180646197698138208358
