| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 4·7-s + 8-s + 9-s − 10-s − 2·11-s + 12-s + 2·13-s + 4·14-s − 15-s + 16-s − 2·17-s + 18-s + 2·19-s − 20-s + 4·21-s − 2·22-s + 24-s + 25-s + 2·26-s + 27-s + 4·28-s + 2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.554·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.872·21-s − 0.426·22-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.755·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.323028531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.323028531\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−9.997711853294076131540262050453, −8.678448890720004138812556129253, −8.175086931274304777032749837299, −7.48044158538181617965029617091, −6.51010129479966833930515773997, −5.25256359120867251767225408448, −4.65757573905330316522778821333, −3.70994421105600557462843234506, −2.60721282477734981997386477165, −1.46077007465007974786279625764,
1.46077007465007974786279625764, 2.60721282477734981997386477165, 3.70994421105600557462843234506, 4.65757573905330316522778821333, 5.25256359120867251767225408448, 6.51010129479966833930515773997, 7.48044158538181617965029617091, 8.175086931274304777032749837299, 8.678448890720004138812556129253, 9.997711853294076131540262050453
