| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s − 6·11-s + 12-s + 14-s + 16-s − 6·17-s − 18-s − 8·19-s − 21-s + 6·22-s − 23-s − 24-s + 27-s − 28-s − 6·29-s − 6·31-s − 32-s − 6·33-s + 6·34-s + 36-s − 8·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.80·11-s + 0.288·12-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 1.83·19-s − 0.218·21-s + 1.27·22-s − 0.208·23-s − 0.204·24-s + 0.192·27-s − 0.188·28-s − 1.11·29-s − 1.07·31-s − 0.176·32-s − 1.04·33-s + 1.02·34-s + 1/6·36-s − 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.79891864820328, −15.53083535547846, −15.03876146252730, −14.53413248500724, −13.56638012985881, −13.32172349697827, −12.68188758031276, −12.44653738496643, −11.27666656092973, −10.98422490789544, −10.36509776367407, −10.05712106595606, −9.185056621618106, −8.759391830977901, −8.323111325546096, −7.699653195063133, −7.096469942491071, −6.595846520896004, −5.837753292145289, −5.146054794066196, −4.394846666095936, −3.642202661679145, −2.872538049085794, −2.156094053900641, −1.813536668135220, 0, 0,
1.813536668135220, 2.156094053900641, 2.872538049085794, 3.642202661679145, 4.394846666095936, 5.146054794066196, 5.837753292145289, 6.595846520896004, 7.096469942491071, 7.699653195063133, 8.323111325546096, 8.759391830977901, 9.185056621618106, 10.05712106595606, 10.36509776367407, 10.98422490789544, 11.27666656092973, 12.44653738496643, 12.68188758031276, 13.32172349697827, 13.56638012985881, 14.53413248500724, 15.03876146252730, 15.53083535547846, 15.79891864820328
