| L(s) = 1 | − 4·7-s + 2·11-s − 13-s + 4·17-s − 2·19-s − 2·23-s − 6·29-s + 8·31-s + 6·37-s − 10·41-s − 4·43-s + 9·49-s − 6·53-s − 6·59-s + 2·61-s − 4·67-s + 12·71-s − 2·73-s − 8·77-s + 8·79-s + 12·83-s − 14·89-s + 4·91-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.603·11-s − 0.277·13-s + 0.970·17-s − 0.458·19-s − 0.417·23-s − 1.11·29-s + 1.43·31-s + 0.986·37-s − 1.56·41-s − 0.609·43-s + 9/7·49-s − 0.824·53-s − 0.781·59-s + 0.256·61-s − 0.488·67-s + 1.42·71-s − 0.234·73-s − 0.911·77-s + 0.900·79-s + 1.31·83-s − 1.48·89-s + 0.419·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.261676062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.261676062\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.38967099236584, −15.01742164112948, −14.39626036381661, −13.71752418650878, −13.35589480716323, −12.67662223403439, −12.28905106038671, −11.76481900224177, −11.15080008988430, −10.26228274601075, −10.00675420846048, −9.425525213539594, −9.002090988095869, −8.099907498823374, −7.714910654976828, −6.730034363878398, −6.533757703893967, −5.900592117470056, −5.190049474892609, −4.381915962216700, −3.635123930629124, −3.212137764544469, −2.429644066020771, −1.480448433088175, −0.4542518365671736,
0.4542518365671736, 1.480448433088175, 2.429644066020771, 3.212137764544469, 3.635123930629124, 4.381915962216700, 5.190049474892609, 5.900592117470056, 6.533757703893967, 6.730034363878398, 7.714910654976828, 8.099907498823374, 9.002090988095869, 9.425525213539594, 10.00675420846048, 10.26228274601075, 11.15080008988430, 11.76481900224177, 12.28905106038671, 12.67662223403439, 13.35589480716323, 13.71752418650878, 14.39626036381661, 15.01742164112948, 15.38967099236584
