| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 12-s + 2·13-s − 14-s − 15-s + 16-s + 17-s − 18-s + 19-s − 20-s + 21-s − 24-s + 25-s − 2·26-s + 27-s + 28-s + 6·29-s + 30-s − 4·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.218·21-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.182·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.56086132737024, −13.93447648987081, −13.54393102497806, −12.82209332664930, −12.44104621691810, −11.73662975269670, −11.41544386557980, −10.87292645839761, −10.24676673787108, −9.900473347664409, −9.245090023805092, −8.696904929881814, −8.296438707954971, −7.934736980444961, −7.281268634031853, −6.821629255497766, −6.256964024942661, −5.476187128874590, −4.928832755649935, −4.166850021081357, −3.608767863817434, −3.007876474265828, −2.368049507029971, −1.551273021633433, −1.029402298152631, 0,
1.029402298152631, 1.551273021633433, 2.368049507029971, 3.007876474265828, 3.608767863817434, 4.166850021081357, 4.928832755649935, 5.476187128874590, 6.256964024942661, 6.821629255497766, 7.281268634031853, 7.934736980444961, 8.296438707954971, 8.696904929881814, 9.245090023805092, 9.900473347664409, 10.24676673787108, 10.87292645839761, 11.41544386557980, 11.73662975269670, 12.44104621691810, 12.82209332664930, 13.54393102497806, 13.93447648987081, 14.56086132737024
