| L(s) = 1 | + 3-s + 9-s − 4·11-s + 13-s − 2·17-s + 4·19-s − 8·23-s + 27-s + 2·29-s − 8·31-s − 4·33-s + 6·37-s + 39-s − 6·41-s − 4·43-s + 8·47-s − 7·49-s − 2·51-s + 6·53-s + 4·57-s + 12·59-s + 2·61-s − 4·67-s − 8·69-s + 6·73-s + 16·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.485·17-s + 0.917·19-s − 1.66·23-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.696·33-s + 0.986·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s + 1.16·47-s − 49-s − 0.280·51-s + 0.824·53-s + 0.529·57-s + 1.56·59-s + 0.256·61-s − 0.488·67-s − 0.963·69-s + 0.702·73-s + 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.63055476026980, −13.83060904657966, −13.56251926189360, −13.19631152458248, −12.50659300229742, −12.11392486316778, −11.43760169495597, −10.96290903469413, −10.36072035602417, −9.907315761041613, −9.480431999219069, −8.805013643674460, −8.208397872066108, −7.955387695651545, −7.285024411079444, −6.837776522954646, −6.017852172246133, −5.540488511869370, −4.972930772308822, −4.279123861515807, −3.638750506335528, −3.150974939017717, −2.270562590320648, −2.005805040929454, −0.9272528490611272, 0,
0.9272528490611272, 2.005805040929454, 2.270562590320648, 3.150974939017717, 3.638750506335528, 4.279123861515807, 4.972930772308822, 5.540488511869370, 6.017852172246133, 6.837776522954646, 7.285024411079444, 7.955387695651545, 8.208397872066108, 8.805013643674460, 9.480431999219069, 9.907315761041613, 10.36072035602417, 10.96290903469413, 11.43760169495597, 12.11392486316778, 12.50659300229742, 13.19631152458248, 13.56251926189360, 13.83060904657966, 14.63055476026980
