| L(s) = 1 | − 1.46·2-s + 0.157·4-s + 3.40·5-s − 1.63·7-s + 2.70·8-s − 4.99·10-s + 4.80·11-s + 2.73·13-s + 2.40·14-s − 4.29·16-s − 7.37·17-s + 5.92·19-s + 0.536·20-s − 7.05·22-s − 5.55·23-s + 6.56·25-s − 4.01·26-s − 0.257·28-s + 1.77·31-s + 0.889·32-s + 10.8·34-s − 5.55·35-s − 2.62·37-s − 8.69·38-s + 9.20·40-s + 0.781·41-s + 1.63·43-s + ⋯ |
| L(s) = 1 | − 1.03·2-s + 0.0788·4-s + 1.52·5-s − 0.617·7-s + 0.956·8-s − 1.57·10-s + 1.44·11-s + 0.757·13-s + 0.641·14-s − 1.07·16-s − 1.78·17-s + 1.35·19-s + 0.119·20-s − 1.50·22-s − 1.15·23-s + 1.31·25-s − 0.787·26-s − 0.0486·28-s + 0.318·31-s + 0.157·32-s + 1.85·34-s − 0.939·35-s − 0.430·37-s − 1.41·38-s + 1.45·40-s + 0.121·41-s + 0.248·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.511934610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.511934610\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 1.46T + 2T^{2} \) |
| 5 | \( 1 - 3.40T + 5T^{2} \) |
| 7 | \( 1 + 1.63T + 7T^{2} \) |
| 11 | \( 1 - 4.80T + 11T^{2} \) |
| 13 | \( 1 - 2.73T + 13T^{2} \) |
| 17 | \( 1 + 7.37T + 17T^{2} \) |
| 19 | \( 1 - 5.92T + 19T^{2} \) |
| 23 | \( 1 + 5.55T + 23T^{2} \) |
| 31 | \( 1 - 1.77T + 31T^{2} \) |
| 37 | \( 1 + 2.62T + 37T^{2} \) |
| 41 | \( 1 - 0.781T + 41T^{2} \) |
| 43 | \( 1 - 1.63T + 43T^{2} \) |
| 47 | \( 1 + 2.40T + 47T^{2} \) |
| 53 | \( 1 - 3.67T + 53T^{2} \) |
| 59 | \( 1 - 2.63T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 5.47T + 67T^{2} \) |
| 71 | \( 1 - 2.63T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 3.96T + 97T^{2} \) |
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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
−8.144540473739539137299364172525, −7.05328965743134709108026842733, −6.56510479639613762161303750800, −6.05874531705621761961821736704, −5.19622547368799808669423895633, −4.27473632845879469897304329357, −3.52457684342147602197913162898, −2.26550939929632383332612723620, −1.64944754312659044851959156745, −0.76371125816242132536997948749,
0.76371125816242132536997948749, 1.64944754312659044851959156745, 2.26550939929632383332612723620, 3.52457684342147602197913162898, 4.27473632845879469897304329357, 5.19622547368799808669423895633, 6.05874531705621761961821736704, 6.56510479639613762161303750800, 7.05328965743134709108026842733, 8.144540473739539137299364172525
