L(s) = 1 | + 0.105·2-s − 3-s − 1.98·4-s − 0.105·6-s + 3.91·7-s − 0.420·8-s + 9-s + 6.01·11-s + 1.98·12-s + 5.26·13-s + 0.412·14-s + 3.93·16-s + 1.81·17-s + 0.105·18-s + 8.19·19-s − 3.91·21-s + 0.634·22-s + 4.74·23-s + 0.420·24-s + 0.555·26-s − 27-s − 7.78·28-s − 4.95·29-s − 5.06·31-s + 1.25·32-s − 6.01·33-s + 0.191·34-s + ⋯ |
L(s) = 1 | + 0.0745·2-s − 0.577·3-s − 0.994·4-s − 0.0430·6-s + 1.47·7-s − 0.148·8-s + 0.333·9-s + 1.81·11-s + 0.574·12-s + 1.46·13-s + 0.110·14-s + 0.983·16-s + 0.439·17-s + 0.0248·18-s + 1.88·19-s − 0.854·21-s + 0.135·22-s + 0.988·23-s + 0.0858·24-s + 0.108·26-s − 0.192·27-s − 1.47·28-s − 0.919·29-s − 0.909·31-s + 0.221·32-s − 1.04·33-s + 0.0327·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.125792761\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.125792761\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 47 | \( 1 + T \) |
good | 2 | \( 1 - 0.105T + 2T^{2} \) |
| 7 | \( 1 - 3.91T + 7T^{2} \) |
| 11 | \( 1 - 6.01T + 11T^{2} \) |
| 13 | \( 1 - 5.26T + 13T^{2} \) |
| 17 | \( 1 - 1.81T + 17T^{2} \) |
| 19 | \( 1 - 8.19T + 19T^{2} \) |
| 23 | \( 1 - 4.74T + 23T^{2} \) |
| 29 | \( 1 + 4.95T + 29T^{2} \) |
| 31 | \( 1 + 5.06T + 31T^{2} \) |
| 37 | \( 1 + 0.310T + 37T^{2} \) |
| 41 | \( 1 - 2.61T + 41T^{2} \) |
| 43 | \( 1 - 4.72T + 43T^{2} \) |
| 53 | \( 1 + 3.05T + 53T^{2} \) |
| 59 | \( 1 + 5.83T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 8.37T + 67T^{2} \) |
| 71 | \( 1 + 9.96T + 71T^{2} \) |
| 73 | \( 1 + 3.65T + 73T^{2} \) |
| 79 | \( 1 - 9.00T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 8.60T + 89T^{2} \) |
| 97 | \( 1 + 8.42T + 97T^{2} \) |
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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
−8.815892212598927205614014996079, −7.79943097196768484164071380155, −7.21870028743273662582989752927, −6.07545986587936660877525636115, −5.50175917270271207039721207297, −4.80341762858625497708017052918, −3.98356314308092729607409205054, −3.42052354607387992939974960062, −1.36798866624388428761808435715, −1.15333095942352962983512436307,
1.15333095942352962983512436307, 1.36798866624388428761808435715, 3.42052354607387992939974960062, 3.98356314308092729607409205054, 4.80341762858625497708017052918, 5.50175917270271207039721207297, 6.07545986587936660877525636115, 7.21870028743273662582989752927, 7.79943097196768484164071380155, 8.815892212598927205614014996079