| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 3·7-s + 8-s − 2·9-s + 10-s + 2·11-s − 12-s − 13-s + 3·14-s − 15-s + 16-s + 3·17-s − 2·18-s + 5·19-s + 20-s − 3·21-s + 2·22-s − 6·23-s − 24-s + 25-s − 26-s + 5·27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s − 0.277·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.471·18-s + 1.14·19-s + 0.223·20-s − 0.654·21-s + 0.426·22-s − 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.962·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−14.91762737268731, −14.32429316539769, −14.08107826721492, −13.68921687186057, −12.99854212220963, −12.09787427707047, −11.97215085724233, −11.60356529313773, −11.01557888281354, −10.42636679205169, −9.874430975626352, −9.342166691011757, −8.463735324461689, −8.075612325702274, −7.507856691497605, −6.709746586110874, −6.264488175219430, −5.636052880113215, −5.060519046379097, −4.905043417241642, −3.934010476836744, −3.307398608933672, −2.619021197706556, −1.659888606158350, −1.313471717975326, 0,
1.313471717975326, 1.659888606158350, 2.619021197706556, 3.307398608933672, 3.934010476836744, 4.905043417241642, 5.060519046379097, 5.636052880113215, 6.264488175219430, 6.709746586110874, 7.507856691497605, 8.075612325702274, 8.463735324461689, 9.342166691011757, 9.874430975626352, 10.42636679205169, 11.01557888281354, 11.60356529313773, 11.97215085724233, 12.09787427707047, 12.99854212220963, 13.68921687186057, 14.08107826721492, 14.32429316539769, 14.91762737268731
