| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s − 2·9-s − 6·11-s − 12-s − 5·13-s − 14-s + 16-s + 3·17-s − 2·18-s + 21-s − 6·22-s + 3·23-s − 24-s − 5·25-s − 5·26-s + 5·27-s − 28-s − 9·29-s + 4·31-s + 32-s + 6·33-s + 3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s − 1.80·11-s − 0.288·12-s − 1.38·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.471·18-s + 0.218·21-s − 1.27·22-s + 0.625·23-s − 0.204·24-s − 25-s − 0.980·26-s + 0.962·27-s − 0.188·28-s − 1.67·29-s + 0.718·31-s + 0.176·32-s + 1.04·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−10.22778802623068126404197182867, −9.303239213765892386789415426985, −7.87022552956968366194501622566, −7.40476303111443747598071967849, −6.09727019472392450239856120896, −5.41480429146700181457700017195, −4.76456060038860011441331940251, −3.24158791522542483628555695808, −2.38005475536401603079615670633, 0,
2.38005475536401603079615670633, 3.24158791522542483628555695808, 4.76456060038860011441331940251, 5.41480429146700181457700017195, 6.09727019472392450239856120896, 7.40476303111443747598071967849, 7.87022552956968366194501622566, 9.303239213765892386789415426985, 10.22778802623068126404197182867
