| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 2·7-s − 8-s + 9-s + 10-s + 6·11-s − 12-s − 13-s − 2·14-s + 15-s + 16-s + 6·17-s − 18-s + 2·19-s − 20-s − 2·21-s − 6·22-s + 24-s + 25-s + 26-s − 27-s + 2·28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s − 0.288·12-s − 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.458·19-s − 0.223·20-s − 0.436·21-s − 1.27·22-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.377·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.206250750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.206250750\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.32687311853220, −11.83113444375430, −11.46951612140217, −11.36829888307956, −10.79516628218508, −10.03711663460883, −9.768272152544715, −9.535513202936531, −8.768711977854292, −8.358303329366586, −7.963167179054011, −7.483788661715508, −7.026287675502058, −6.518869677052988, −6.102508531954685, −5.624438158999585, −4.899428506924324, −4.561088394621271, −4.025119420095232, −3.307754726019852, −3.018202962214131, −2.049245567447400, −1.386238845016317, −1.139948366898348, −0.5155238739126968,
0.5155238739126968, 1.139948366898348, 1.386238845016317, 2.049245567447400, 3.018202962214131, 3.307754726019852, 4.025119420095232, 4.561088394621271, 4.899428506924324, 5.624438158999585, 6.102508531954685, 6.518869677052988, 7.026287675502058, 7.483788661715508, 7.963167179054011, 8.358303329366586, 8.768711977854292, 9.535513202936531, 9.768272152544715, 10.03711663460883, 10.79516628218508, 11.36829888307956, 11.46951612140217, 11.83113444375430, 12.32687311853220
