L(s) = 1 | − 2-s − 3-s + 4-s + 4·5-s + 6-s + 7-s − 8-s + 9-s − 4·10-s − 3·11-s − 12-s − 14-s − 4·15-s + 16-s − 5·17-s − 18-s + 3·19-s + 4·20-s − 21-s + 3·22-s + 6·23-s + 24-s + 11·25-s − 27-s + 28-s − 9·29-s + 4·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.904·11-s − 0.288·12-s − 0.267·14-s − 1.03·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s + 0.688·19-s + 0.894·20-s − 0.218·21-s + 0.639·22-s + 1.25·23-s + 0.204·24-s + 11/5·25-s − 0.192·27-s + 0.188·28-s − 1.67·29-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.630390153\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.630390153\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.86658573123638376381852450334, −7.21866406789026642455443593172, −6.49875818886037998251216400759, −5.88530162756376128209565188532, −5.24272266713152856730530480923, −4.75810787980735068139739354523, −3.31251826490877898940880451216, −2.28747649194683305451719810594, −1.83761024782768121265174655187, −0.74312015001095226265609756885,
0.74312015001095226265609756885, 1.83761024782768121265174655187, 2.28747649194683305451719810594, 3.31251826490877898940880451216, 4.75810787980735068139739354523, 5.24272266713152856730530480923, 5.88530162756376128209565188532, 6.49875818886037998251216400759, 7.21866406789026642455443593172, 7.86658573123638376381852450334