L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 5·7-s + 9-s + 11-s + 2·12-s − 2·13-s + 10·14-s − 4·16-s + 17-s + 2·18-s − 19-s + 5·21-s + 2·22-s + 4·23-s − 4·26-s + 27-s + 10·28-s − 2·29-s − 6·31-s − 8·32-s + 33-s + 2·34-s + 2·36-s − 2·38-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 1.88·7-s + 1/3·9-s + 0.301·11-s + 0.577·12-s − 0.554·13-s + 2.67·14-s − 16-s + 0.242·17-s + 0.471·18-s − 0.229·19-s + 1.09·21-s + 0.426·22-s + 0.834·23-s − 0.784·26-s + 0.192·27-s + 1.88·28-s − 0.371·29-s − 1.07·31-s − 1.41·32-s + 0.174·33-s + 0.342·34-s + 1/3·36-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.968994303\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.968994303\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−9.341053920747992022463934553266, −8.723409361965311529712840067335, −7.74191106110524052069368926798, −7.15886697828319609358757774448, −5.94366419557132992161838079855, −5.07072622007111059188732569210, −4.55783516638495121923212657411, −3.69330683704517818549980694450, −2.57494569661476323183620903482, −1.60475124369086001622436622243,
1.60475124369086001622436622243, 2.57494569661476323183620903482, 3.69330683704517818549980694450, 4.55783516638495121923212657411, 5.07072622007111059188732569210, 5.94366419557132992161838079855, 7.15886697828319609358757774448, 7.74191106110524052069368926798, 8.723409361965311529712840067335, 9.341053920747992022463934553266