| L(s) = 1 | − 0.539·3-s − 5-s + 0.630·7-s − 2.70·9-s + 1.70·11-s + 3.17·13-s + 0.539·15-s + 1.41·17-s − 19-s − 0.340·21-s − 4.04·23-s + 25-s + 3.07·27-s − 3.75·29-s + 1.41·31-s − 0.921·33-s − 0.630·35-s + 0.986·37-s − 1.70·39-s − 9.26·41-s − 5.70·43-s + 2.70·45-s − 4.04·47-s − 6.60·49-s − 0.764·51-s + 7.32·53-s − 1.70·55-s + ⋯ |
| L(s) = 1 | − 0.311·3-s − 0.447·5-s + 0.238·7-s − 0.903·9-s + 0.515·11-s + 0.879·13-s + 0.139·15-s + 0.344·17-s − 0.229·19-s − 0.0742·21-s − 0.844·23-s + 0.200·25-s + 0.592·27-s − 0.697·29-s + 0.254·31-s − 0.160·33-s − 0.106·35-s + 0.162·37-s − 0.273·39-s − 1.44·41-s − 0.870·43-s + 0.403·45-s − 0.590·47-s − 0.943·49-s − 0.107·51-s + 1.00·53-s − 0.230·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.539T + 3T^{2} \) |
| 7 | \( 1 - 0.630T + 7T^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 13 | \( 1 - 3.17T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 23 | \( 1 + 4.04T + 23T^{2} \) |
| 29 | \( 1 + 3.75T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 - 0.986T + 37T^{2} \) |
| 41 | \( 1 + 9.26T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 + 4.04T + 47T^{2} \) |
| 53 | \( 1 - 7.32T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 + 6.14T + 67T^{2} \) |
| 71 | \( 1 - 7.94T + 71T^{2} \) |
| 73 | \( 1 + 9.91T + 73T^{2} \) |
| 79 | \( 1 - 5.02T + 79T^{2} \) |
| 83 | \( 1 + 3.86T + 83T^{2} \) |
| 89 | \( 1 - 5.60T + 89T^{2} \) |
| 97 | \( 1 + 0.275T + 97T^{2} \) |
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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
−7.87488141455928989374984649605, −6.87372448087659206438027077263, −6.32602882600973618583049577412, −5.56982257594971124084840958331, −4.93138067550564493788895576881, −3.88111850941777763420051327814, −3.43781554497291866041327921215, −2.29483547525440025268823915940, −1.22886988820728869147530092022, 0,
1.22886988820728869147530092022, 2.29483547525440025268823915940, 3.43781554497291866041327921215, 3.88111850941777763420051327814, 4.93138067550564493788895576881, 5.56982257594971124084840958331, 6.32602882600973618583049577412, 6.87372448087659206438027077263, 7.87488141455928989374984649605
