| L(s) = 1 | − 3.23·5-s − 7-s + 4.47·11-s + 4.47·13-s + 3.23·17-s + 19-s + 8.47·23-s + 5.47·25-s + 2.76·29-s − 2.47·31-s + 3.23·35-s + 4.47·37-s − 2·41-s + 8·43-s + 1.23·47-s + 49-s + 1.23·53-s − 14.4·55-s − 4·59-s − 12.4·61-s − 14.4·65-s − 10.4·67-s + 0.763·71-s − 4.47·73-s − 4.47·77-s − 12·79-s + 7.70·83-s + ⋯ |
| L(s) = 1 | − 1.44·5-s − 0.377·7-s + 1.34·11-s + 1.24·13-s + 0.784·17-s + 0.229·19-s + 1.76·23-s + 1.09·25-s + 0.513·29-s − 0.444·31-s + 0.546·35-s + 0.735·37-s − 0.312·41-s + 1.21·43-s + 0.180·47-s + 0.142·49-s + 0.169·53-s − 1.95·55-s − 0.520·59-s − 1.59·61-s − 1.79·65-s − 1.27·67-s + 0.0906·71-s − 0.523·73-s − 0.509·77-s − 1.35·79-s + 0.846·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.899691664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.899691664\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 3.23T + 17T^{2} \) |
| 23 | \( 1 - 8.47T + 23T^{2} \) |
| 29 | \( 1 - 2.76T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 1.23T + 47T^{2} \) |
| 53 | \( 1 - 1.23T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 0.763T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 7.70T + 83T^{2} \) |
| 89 | \( 1 + 6.94T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−7.52823803322846159759648154169, −7.18889788851027380560146848903, −6.34575278849340594344900133085, −5.80057983565297474381585157848, −4.68493701240582109385200006039, −4.13135328882348988959716002594, −3.41787713104582311019878868858, −3.01659255582841408667145002889, −1.40009131734888899170382690652, −0.74511465595496916241196092036,
0.74511465595496916241196092036, 1.40009131734888899170382690652, 3.01659255582841408667145002889, 3.41787713104582311019878868858, 4.13135328882348988959716002594, 4.68493701240582109385200006039, 5.80057983565297474381585157848, 6.34575278849340594344900133085, 7.18889788851027380560146848903, 7.52823803322846159759648154169
