| L(s) = 1 | − 1.56·3-s − 3.56·5-s − 0.561·9-s + 11-s + 5.12·13-s + 5.56·15-s − 2·17-s − 4·19-s − 2.43·23-s + 7.68·25-s + 5.56·27-s − 5.12·29-s − 5.56·31-s − 1.56·33-s − 7.56·37-s − 8·39-s + 1.12·41-s + 7.12·43-s + 2·45-s + 8·47-s + 3.12·51-s + 12.2·53-s − 3.56·55-s + 6.24·57-s + 7.80·59-s − 1.12·61-s − 18.2·65-s + ⋯ |
| L(s) = 1 | − 0.901·3-s − 1.59·5-s − 0.187·9-s + 0.301·11-s + 1.42·13-s + 1.43·15-s − 0.485·17-s − 0.917·19-s − 0.508·23-s + 1.53·25-s + 1.07·27-s − 0.951·29-s − 0.998·31-s − 0.271·33-s − 1.24·37-s − 1.28·39-s + 0.175·41-s + 1.08·43-s + 0.298·45-s + 1.16·47-s + 0.437·51-s + 1.68·53-s − 0.480·55-s + 0.827·57-s + 1.01·59-s − 0.143·61-s − 2.26·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 + 5.12T + 29T^{2} \) |
| 31 | \( 1 + 5.56T + 31T^{2} \) |
| 37 | \( 1 + 7.56T + 37T^{2} \) |
| 41 | \( 1 - 1.12T + 41T^{2} \) |
| 43 | \( 1 - 7.12T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 7.80T + 59T^{2} \) |
| 61 | \( 1 + 1.12T + 61T^{2} \) |
| 67 | \( 1 + 9.56T + 67T^{2} \) |
| 71 | \( 1 - 8.68T + 71T^{2} \) |
| 73 | \( 1 + 5.12T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 0.876T + 83T^{2} \) |
| 89 | \( 1 + 2.68T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.29985050209686827612658759846, −6.81359986000869526084346729084, −5.96563997839194957250398797818, −5.51158118552307337121038958690, −4.44993997262429244241923068084, −3.93602802262462089723564526126, −3.41041385135408346467192367714, −2.14224078262278293648941240445, −0.866503153579054288206624202194, 0,
0.866503153579054288206624202194, 2.14224078262278293648941240445, 3.41041385135408346467192367714, 3.93602802262462089723564526126, 4.44993997262429244241923068084, 5.51158118552307337121038958690, 5.96563997839194957250398797818, 6.81359986000869526084346729084, 7.29985050209686827612658759846
