| L(s) = 1 | + 2.74·2-s − 3.04·3-s + 5.51·4-s − 5-s − 8.33·6-s − 7-s + 9.63·8-s + 6.24·9-s − 2.74·10-s − 16.7·12-s + 6.59·13-s − 2.74·14-s + 3.04·15-s + 15.3·16-s − 4.86·17-s + 17.1·18-s + 0.907·19-s − 5.51·20-s + 3.04·21-s + 3.01·23-s − 29.3·24-s + 25-s + 18.0·26-s − 9.87·27-s − 5.51·28-s − 0.914·29-s + 8.33·30-s + ⋯ |
| L(s) = 1 | + 1.93·2-s − 1.75·3-s + 2.75·4-s − 0.447·5-s − 3.40·6-s − 0.377·7-s + 3.40·8-s + 2.08·9-s − 0.866·10-s − 4.84·12-s + 1.82·13-s − 0.732·14-s + 0.785·15-s + 3.84·16-s − 1.17·17-s + 4.03·18-s + 0.208·19-s − 1.23·20-s + 0.663·21-s + 0.628·23-s − 5.98·24-s + 0.200·25-s + 3.54·26-s − 1.90·27-s − 1.04·28-s − 0.169·29-s + 1.52·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.880875504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.880875504\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 3 | \( 1 + 3.04T + 3T^{2} \) |
| 13 | \( 1 - 6.59T + 13T^{2} \) |
| 17 | \( 1 + 4.86T + 17T^{2} \) |
| 19 | \( 1 - 0.907T + 19T^{2} \) |
| 23 | \( 1 - 3.01T + 23T^{2} \) |
| 29 | \( 1 + 0.914T + 29T^{2} \) |
| 31 | \( 1 + 5.33T + 31T^{2} \) |
| 37 | \( 1 + 2.76T + 37T^{2} \) |
| 41 | \( 1 + 5.06T + 41T^{2} \) |
| 43 | \( 1 - 1.67T + 43T^{2} \) |
| 47 | \( 1 - 9.45T + 47T^{2} \) |
| 53 | \( 1 + 3.38T + 53T^{2} \) |
| 59 | \( 1 + 1.13T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 7.50T + 67T^{2} \) |
| 71 | \( 1 - 3.74T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 7.76T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−7.989037880270648994869584988150, −6.91337741688229784190321114310, −6.68583556084670271007665777689, −6.02264394076570788869662173632, −5.37718893633503701854534933928, −4.81399139597817174136231920660, −3.89917753734479293685986716280, −3.55364622560951794528543635160, −2.10353132311963614335838832748, −0.941644749399451622078077557137,
0.941644749399451622078077557137, 2.10353132311963614335838832748, 3.55364622560951794528543635160, 3.89917753734479293685986716280, 4.81399139597817174136231920660, 5.37718893633503701854534933928, 6.02264394076570788869662173632, 6.68583556084670271007665777689, 6.91337741688229784190321114310, 7.989037880270648994869584988150
