| L(s) = 1 | + 2-s − 4-s + 5-s − 4·7-s − 3·8-s + 10-s − 4·14-s − 16-s + 4·17-s − 4·19-s − 20-s + 8·23-s + 25-s + 4·28-s + 8·29-s + 4·31-s + 5·32-s + 4·34-s − 4·35-s − 4·37-s − 4·38-s − 3·40-s − 2·41-s − 8·43-s + 8·46-s − 8·47-s + 9·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.51·7-s − 1.06·8-s + 0.316·10-s − 1.06·14-s − 1/4·16-s + 0.970·17-s − 0.917·19-s − 0.223·20-s + 1.66·23-s + 1/5·25-s + 0.755·28-s + 1.48·29-s + 0.718·31-s + 0.883·32-s + 0.685·34-s − 0.676·35-s − 0.657·37-s − 0.648·38-s − 0.474·40-s − 0.312·41-s − 1.21·43-s + 1.17·46-s − 1.16·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−7.29497272181629200871838036262, −6.51229138252165770500062445454, −6.22739087506708215981991892218, −5.32822189024860931198520473817, −4.78697608655564783868843775468, −3.88351610641135160380934689818, −3.09901814197336710134773842184, −2.75302751382602634483772277568, −1.18101632128069620631375397267, 0,
1.18101632128069620631375397267, 2.75302751382602634483772277568, 3.09901814197336710134773842184, 3.88351610641135160380934689818, 4.78697608655564783868843775468, 5.32822189024860931198520473817, 6.22739087506708215981991892218, 6.51229138252165770500062445454, 7.29497272181629200871838036262
