| L(s) = 1 | − 2.61·3-s + 1.61·5-s + 3.85·7-s + 3.85·9-s − 2.38·13-s − 4.23·15-s + 2.38·17-s + 3.85·19-s − 10.0·21-s + 2.47·23-s − 2.38·25-s − 2.23·27-s − 8.61·29-s − 0.854·31-s + 6.23·35-s + 1.85·37-s + 6.23·39-s + 8.61·41-s + 6.23·45-s − 1.38·47-s + 7.85·49-s − 6.23·51-s + 4.09·53-s − 10.0·57-s + 1.09·59-s − 2.38·61-s + 14.8·63-s + ⋯ |
| L(s) = 1 | − 1.51·3-s + 0.723·5-s + 1.45·7-s + 1.28·9-s − 0.660·13-s − 1.09·15-s + 0.577·17-s + 0.884·19-s − 2.20·21-s + 0.515·23-s − 0.476·25-s − 0.430·27-s − 1.60·29-s − 0.153·31-s + 1.05·35-s + 0.304·37-s + 0.998·39-s + 1.34·41-s + 0.929·45-s − 0.201·47-s + 1.12·49-s − 0.873·51-s + 0.561·53-s − 1.33·57-s + 0.141·59-s − 0.304·61-s + 1.87·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.661497777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.661497777\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 13 | \( 1 + 2.38T + 13T^{2} \) |
| 17 | \( 1 - 2.38T + 17T^{2} \) |
| 19 | \( 1 - 3.85T + 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 + 8.61T + 29T^{2} \) |
| 31 | \( 1 + 0.854T + 31T^{2} \) |
| 37 | \( 1 - 1.85T + 37T^{2} \) |
| 41 | \( 1 - 8.61T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 1.38T + 47T^{2} \) |
| 53 | \( 1 - 4.09T + 53T^{2} \) |
| 59 | \( 1 - 1.09T + 59T^{2} \) |
| 61 | \( 1 + 2.38T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 6.38T + 71T^{2} \) |
| 73 | \( 1 + 0.909T + 73T^{2} \) |
| 79 | \( 1 - 7.14T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 0.472T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−7.60462485246640492582457430941, −7.23452566071152088462777998802, −6.25944528584449015811712155721, −5.55556714598172392464477634830, −5.27818397092439376340739702189, −4.67064390476393549825214061209, −3.74746394639842196511930865297, −2.40171843338407706661653947189, −1.58119383557012001890798133815, −0.74101033350946798468464817260,
0.74101033350946798468464817260, 1.58119383557012001890798133815, 2.40171843338407706661653947189, 3.74746394639842196511930865297, 4.67064390476393549825214061209, 5.27818397092439376340739702189, 5.55556714598172392464477634830, 6.25944528584449015811712155721, 7.23452566071152088462777998802, 7.60462485246640492582457430941
