| L(s) = 1 | + 1.41·3-s + 5-s + 7-s − 0.999·9-s − 11-s − 0.585·13-s + 1.41·15-s − 6.24·17-s − 5.65·19-s + 1.41·21-s − 2·23-s + 25-s − 5.65·27-s − 8.82·29-s − 7.41·31-s − 1.41·33-s + 35-s + 2.82·37-s − 0.828·39-s − 4.24·41-s − 2·43-s − 0.999·45-s + 0.242·47-s + 49-s − 8.82·51-s + 12.4·53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.816·3-s + 0.447·5-s + 0.377·7-s − 0.333·9-s − 0.301·11-s − 0.162·13-s + 0.365·15-s − 1.51·17-s − 1.29·19-s + 0.308·21-s − 0.417·23-s + 0.200·25-s − 1.08·27-s − 1.63·29-s − 1.33·31-s − 0.246·33-s + 0.169·35-s + 0.464·37-s − 0.132·39-s − 0.662·41-s − 0.304·43-s − 0.149·45-s + 0.0353·47-s + 0.142·49-s − 1.23·51-s + 1.71·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 13 | \( 1 + 0.585T + 13T^{2} \) |
| 17 | \( 1 + 6.24T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 8.82T + 29T^{2} \) |
| 31 | \( 1 + 7.41T + 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 0.242T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 7.41T + 59T^{2} \) |
| 61 | \( 1 + 3.75T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 1.75T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 6.34T + 83T^{2} \) |
| 89 | \( 1 + 8.82T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−8.499931861775439164885546415871, −7.69131373491709364360975261962, −6.91908120827966978170678230568, −6.02467134476804851231772005135, −5.29008571838529932540169496806, −4.29864597227468976325666269376, −3.53200582381120166031156383816, −2.27164307413442918325971123258, −2.01833202017254655693593127031, 0,
2.01833202017254655693593127031, 2.27164307413442918325971123258, 3.53200582381120166031156383816, 4.29864597227468976325666269376, 5.29008571838529932540169496806, 6.02467134476804851231772005135, 6.91908120827966978170678230568, 7.69131373491709364360975261962, 8.499931861775439164885546415871
