L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·5-s − 2·6-s + 4·7-s + 9-s − 4·10-s − 3·11-s − 2·12-s + 8·14-s + 2·15-s − 4·16-s + 17-s + 2·18-s − 7·19-s − 4·20-s − 4·21-s − 6·22-s − 25-s − 27-s + 8·28-s + 9·29-s + 4·30-s + 8·31-s − 8·32-s + 3·33-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.894·5-s − 0.816·6-s + 1.51·7-s + 1/3·9-s − 1.26·10-s − 0.904·11-s − 0.577·12-s + 2.13·14-s + 0.516·15-s − 16-s + 0.242·17-s + 0.471·18-s − 1.60·19-s − 0.894·20-s − 0.872·21-s − 1.27·22-s − 1/5·25-s − 0.192·27-s + 1.51·28-s + 1.67·29-s + 0.730·30-s + 1.43·31-s − 1.41·32-s + 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.940650029\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.940650029\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.70224547312158870414838139451, −6.88644089951924194845284458992, −6.20963882703000360047142037294, −5.46413829867624494532195544092, −4.79703434369813196275546290927, −4.46486231340171915326858035439, −3.86265505185246303124129992521, −2.77462754106791638079684385007, −2.05406424872503173743387964745, −0.67993303887674977063741634232,
0.67993303887674977063741634232, 2.05406424872503173743387964745, 2.77462754106791638079684385007, 3.86265505185246303124129992521, 4.46486231340171915326858035439, 4.79703434369813196275546290927, 5.46413829867624494532195544092, 6.20963882703000360047142037294, 6.88644089951924194845284458992, 7.70224547312158870414838139451