| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 3.53·7-s − 8-s + 9-s − 1.53·11-s − 12-s − 6·13-s + 3.53·14-s + 16-s + 4·17-s − 18-s − 3.53·19-s + 3.53·21-s + 1.53·22-s + 1.53·23-s + 24-s + 6·26-s − 27-s − 3.53·28-s + 31-s − 32-s + 1.53·33-s − 4·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.33·7-s − 0.353·8-s + 0.333·9-s − 0.461·11-s − 0.288·12-s − 1.66·13-s + 0.943·14-s + 0.250·16-s + 0.970·17-s − 0.235·18-s − 0.810·19-s + 0.770·21-s + 0.326·22-s + 0.319·23-s + 0.204·24-s + 1.17·26-s − 0.192·27-s − 0.667·28-s + 0.179·31-s − 0.176·32-s + 0.266·33-s − 0.685·34-s + 0.166·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2983485706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2983485706\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 3.53T + 7T^{2} \) |
| 11 | \( 1 + 1.53T + 11T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 3.53T + 19T^{2} \) |
| 23 | \( 1 - 1.53T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 37 | \( 1 + 9.06T + 37T^{2} \) |
| 41 | \( 1 + 9.06T + 41T^{2} \) |
| 43 | \( 1 - 0.468T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 5.53T + 53T^{2} \) |
| 59 | \( 1 - 7.06T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 4.46T + 71T^{2} \) |
| 73 | \( 1 - 0.468T + 73T^{2} \) |
| 79 | \( 1 + 0.468T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 - 1.53T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−8.268447229500425825361775877490, −7.53222033303086431425640823331, −6.82573054175585324988429718743, −6.39776535727201225040594457088, −5.40398441213482391157805645112, −4.82941021711112342861154095291, −3.53793292344824818172434057768, −2.84830918086831138604207225624, −1.81305907596016115336129655449, −0.32954572545734250328408348105,
0.32954572545734250328408348105, 1.81305907596016115336129655449, 2.84830918086831138604207225624, 3.53793292344824818172434057768, 4.82941021711112342861154095291, 5.40398441213482391157805645112, 6.39776535727201225040594457088, 6.82573054175585324988429718743, 7.53222033303086431425640823331, 8.268447229500425825361775877490
