| L(s) = 1 | − 2-s + 2.28·3-s + 4-s − 5-s − 2.28·6-s − 1.23·7-s − 8-s + 2.23·9-s + 10-s + 1.95·11-s + 2.28·12-s − 0.874·13-s + 1.23·14-s − 2.28·15-s + 16-s + 6.32·17-s − 2.23·18-s − 2.76·19-s − 20-s − 2.82·21-s − 1.95·22-s − 3.16·23-s − 2.28·24-s + 25-s + 0.874·26-s − 1.74·27-s − 1.23·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.32·3-s + 0.5·4-s − 0.447·5-s − 0.934·6-s − 0.467·7-s − 0.353·8-s + 0.745·9-s + 0.316·10-s + 0.589·11-s + 0.660·12-s − 0.242·13-s + 0.330·14-s − 0.590·15-s + 0.250·16-s + 1.53·17-s − 0.527·18-s − 0.634·19-s − 0.223·20-s − 0.617·21-s − 0.416·22-s − 0.659·23-s − 0.467·24-s + 0.200·25-s + 0.171·26-s − 0.336·27-s − 0.233·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.104624223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.104624223\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 - 2.28T + 3T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 - 1.95T + 11T^{2} \) |
| 13 | \( 1 + 0.874T + 13T^{2} \) |
| 17 | \( 1 - 6.32T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 - 6.86T + 29T^{2} \) |
| 37 | \( 1 + 0.874T + 37T^{2} \) |
| 41 | \( 1 + 1.23T + 41T^{2} \) |
| 43 | \( 1 - 9.69T + 43T^{2} \) |
| 47 | \( 1 - 1.70T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 3.70T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 0.763T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 2.95T + 83T^{2} \) |
| 89 | \( 1 - 3.57T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−7.896895953905557748334841924972, −7.30185439221329288255183511433, −6.51295086038913962638315405964, −5.88312682765922129578987255439, −4.77218630312413135903433983445, −3.84007104320514662068717778288, −3.32485067809093110023620106391, −2.63440649070297370809374326545, −1.78432389997571686180961675643, −0.72021458917826176058807563263,
0.72021458917826176058807563263, 1.78432389997571686180961675643, 2.63440649070297370809374326545, 3.32485067809093110023620106391, 3.84007104320514662068717778288, 4.77218630312413135903433983445, 5.88312682765922129578987255439, 6.51295086038913962638315405964, 7.30185439221329288255183511433, 7.896895953905557748334841924972
