| L(s) = 1 | + 2-s − 3·3-s + 4-s − 3·6-s − 7-s + 8-s + 6·9-s − 5·11-s − 3·12-s − 6·13-s − 14-s + 16-s − 17-s + 6·18-s − 3·19-s + 3·21-s − 5·22-s − 3·24-s − 6·26-s − 9·27-s − 28-s − 6·29-s − 4·31-s + 32-s + 15·33-s − 34-s + 6·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.377·7-s + 0.353·8-s + 2·9-s − 1.50·11-s − 0.866·12-s − 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 1.41·18-s − 0.688·19-s + 0.654·21-s − 1.06·22-s − 0.612·24-s − 1.17·26-s − 1.73·27-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 2.61·33-s − 0.171·34-s + 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 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
−11.11858258197447904087693203763, −10.45494407305386764464659959764, −9.627920882288908715420031969238, −7.70054391482804408966188304160, −6.95538626651508714624591948210, −5.84573096199937208498827543184, −5.20618442524944639616146822933, −4.32124854619714589271574708665, −2.44905781159727834481891285350, 0,
2.44905781159727834481891285350, 4.32124854619714589271574708665, 5.20618442524944639616146822933, 5.84573096199937208498827543184, 6.95538626651508714624591948210, 7.70054391482804408966188304160, 9.627920882288908715420031969238, 10.45494407305386764464659959764, 11.11858258197447904087693203763
