| L(s) = 1 | + 2·5-s + 7-s − 4·13-s + 6·17-s + 2·19-s + 2·23-s − 25-s + 2·29-s − 8·31-s + 2·35-s − 2·37-s − 6·41-s − 2·43-s − 10·47-s + 49-s + 14·53-s − 12·61-s − 8·65-s + 8·67-s + 6·71-s − 6·73-s + 4·79-s − 4·83-s + 12·85-s + 12·89-s − 4·91-s + 4·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 1.10·13-s + 1.45·17-s + 0.458·19-s + 0.417·23-s − 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.338·35-s − 0.328·37-s − 0.937·41-s − 0.304·43-s − 1.45·47-s + 1/7·49-s + 1.92·53-s − 1.53·61-s − 0.992·65-s + 0.977·67-s + 0.712·71-s − 0.702·73-s + 0.450·79-s − 0.439·83-s + 1.30·85-s + 1.27·89-s − 0.419·91-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.83126365763811, −13.27733163713962, −12.93986775563850, −12.25033072625798, −11.88639900043994, −11.56231318249591, −10.65050072844771, −10.43265584783417, −9.827558292434163, −9.511060457217731, −9.046235814167546, −8.336201713915746, −7.848878782927293, −7.346165493495328, −6.909094361623593, −6.242476979739490, −5.641389363010210, −5.096904636596857, −5.043537978563459, −4.035456499633209, −3.419056078635542, −2.889317679954381, −2.136418914331603, −1.671231210267846, −0.9840330072506921, 0,
0.9840330072506921, 1.671231210267846, 2.136418914331603, 2.889317679954381, 3.419056078635542, 4.035456499633209, 5.043537978563459, 5.096904636596857, 5.641389363010210, 6.242476979739490, 6.909094361623593, 7.346165493495328, 7.848878782927293, 8.336201713915746, 9.046235814167546, 9.511060457217731, 9.827558292434163, 10.43265584783417, 10.65050072844771, 11.56231318249591, 11.88639900043994, 12.25033072625798, 12.93986775563850, 13.27733163713962, 13.83126365763811
