| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 5·7-s − 8-s + 9-s − 6·11-s + 12-s + 2·13-s − 5·14-s + 16-s − 17-s − 18-s − 19-s + 5·21-s + 6·22-s − 9·23-s − 24-s − 5·25-s − 2·26-s + 27-s + 5·28-s + 6·29-s + 2·31-s − 32-s − 6·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.88·7-s − 0.353·8-s + 1/3·9-s − 1.80·11-s + 0.288·12-s + 0.554·13-s − 1.33·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.229·19-s + 1.09·21-s + 1.27·22-s − 1.87·23-s − 0.204·24-s − 25-s − 0.392·26-s + 0.192·27-s + 0.944·28-s + 1.11·29-s + 0.359·31-s − 0.176·32-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.924716295141091652641160039564, −7.46060105548026587700696725096, −6.42237419695144003273746080829, −5.49567179745016790352397340923, −4.88185337421921511656665885915, −4.06970591650444067727575977115, −2.97032738307948241561194308131, −2.03050870626327678152847331463, −1.60459865728141598540094349752, 0,
1.60459865728141598540094349752, 2.03050870626327678152847331463, 2.97032738307948241561194308131, 4.06970591650444067727575977115, 4.88185337421921511656665885915, 5.49567179745016790352397340923, 6.42237419695144003273746080829, 7.46060105548026587700696725096, 7.924716295141091652641160039564
