| L(s) = 1 | − 2.65·3-s − 5-s + 7-s + 4.05·9-s + 11-s − 2.65·13-s + 2.65·15-s − 5.44·17-s + 4.10·19-s − 2.65·21-s + 0.259·23-s + 25-s − 2.79·27-s − 2·29-s + 4.91·31-s − 2.65·33-s − 35-s − 2.25·37-s + 7.05·39-s − 1.08·41-s + 4.53·43-s − 4.05·45-s − 4.13·47-s + 49-s + 14.4·51-s − 4.36·53-s − 55-s + ⋯ |
| L(s) = 1 | − 1.53·3-s − 0.447·5-s + 0.377·7-s + 1.35·9-s + 0.301·11-s − 0.736·13-s + 0.685·15-s − 1.32·17-s + 0.941·19-s − 0.579·21-s + 0.0541·23-s + 0.200·25-s − 0.537·27-s − 0.371·29-s + 0.882·31-s − 0.462·33-s − 0.169·35-s − 0.371·37-s + 1.12·39-s − 0.169·41-s + 0.691·43-s − 0.603·45-s − 0.603·47-s + 0.142·49-s + 2.02·51-s − 0.599·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2.65T + 3T^{2} \) |
| 13 | \( 1 + 2.65T + 13T^{2} \) |
| 17 | \( 1 + 5.44T + 17T^{2} \) |
| 19 | \( 1 - 4.10T + 19T^{2} \) |
| 23 | \( 1 - 0.259T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4.91T + 31T^{2} \) |
| 37 | \( 1 + 2.25T + 37T^{2} \) |
| 41 | \( 1 + 1.08T + 41T^{2} \) |
| 43 | \( 1 - 4.53T + 43T^{2} \) |
| 47 | \( 1 + 4.13T + 47T^{2} \) |
| 53 | \( 1 + 4.36T + 53T^{2} \) |
| 59 | \( 1 + 5.97T + 59T^{2} \) |
| 61 | \( 1 - 3.80T + 61T^{2} \) |
| 67 | \( 1 + 7.15T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 + 2.55T + 73T^{2} \) |
| 79 | \( 1 - 3.63T + 79T^{2} \) |
| 83 | \( 1 + 6.79T + 83T^{2} \) |
| 89 | \( 1 - 5.31T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−7.49379789510451122581915965355, −6.90439910574245381054672512754, −6.29481968872842863479634363187, −5.52963934089536074915408746457, −4.77803431737844659170523619284, −4.44434050789582462950446322638, −3.33282062092290517319335201682, −2.15206026523181725862382305778, −1.00993013567478273050309231425, 0,
1.00993013567478273050309231425, 2.15206026523181725862382305778, 3.33282062092290517319335201682, 4.44434050789582462950446322638, 4.77803431737844659170523619284, 5.52963934089536074915408746457, 6.29481968872842863479634363187, 6.90439910574245381054672512754, 7.49379789510451122581915965355
