| L(s) = 1 | − 5·5-s + 4·7-s + 12·11-s − 58·13-s − 66·17-s + 100·19-s + 132·23-s + 25·25-s + 90·29-s − 152·31-s − 20·35-s − 34·37-s + 438·41-s − 32·43-s − 204·47-s − 327·49-s − 222·53-s − 60·55-s + 420·59-s + 902·61-s + 290·65-s + 1.02e3·67-s + 432·71-s + 362·73-s + 48·77-s + 160·79-s + 72·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.215·7-s + 0.328·11-s − 1.23·13-s − 0.941·17-s + 1.20·19-s + 1.19·23-s + 1/5·25-s + 0.576·29-s − 0.880·31-s − 0.0965·35-s − 0.151·37-s + 1.66·41-s − 0.113·43-s − 0.633·47-s − 0.953·49-s − 0.575·53-s − 0.147·55-s + 0.926·59-s + 1.89·61-s + 0.553·65-s + 1.86·67-s + 0.722·71-s + 0.580·73-s + 0.0710·77-s + 0.227·79-s + 0.0952·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.698266296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.698266296\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 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
−9.872978703856461894043015861411, −9.239395075058080752134280715308, −8.258302383397885714975232606182, −7.33609288230875640139932095238, −6.71431311286194958059600056474, −5.32884233645066119963398392418, −4.61588179213245866675192192296, −3.43183326484512113881131974588, −2.27736882290125384233412313843, −0.74322804685502860806979532349,
0.74322804685502860806979532349, 2.27736882290125384233412313843, 3.43183326484512113881131974588, 4.61588179213245866675192192296, 5.32884233645066119963398392418, 6.71431311286194958059600056474, 7.33609288230875640139932095238, 8.258302383397885714975232606182, 9.239395075058080752134280715308, 9.872978703856461894043015861411
