| L(s) = 1 | + 2-s + 4-s − 4.17·5-s + 2.28·7-s + 8-s − 4.17·10-s + 4.52·11-s + 2.72·13-s + 2.28·14-s + 16-s + 0.253·19-s − 4.17·20-s + 4.52·22-s − 1.41·23-s + 12.4·25-s + 2.72·26-s + 2.28·28-s − 10.0·29-s − 0.944·31-s + 32-s − 9.53·35-s + 2.48·37-s + 0.253·38-s − 4.17·40-s + 2.67·41-s + 1.29·43-s + 4.52·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.86·5-s + 0.862·7-s + 0.353·8-s − 1.32·10-s + 1.36·11-s + 0.757·13-s + 0.609·14-s + 0.250·16-s + 0.0582·19-s − 0.934·20-s + 0.964·22-s − 0.294·23-s + 2.49·25-s + 0.535·26-s + 0.431·28-s − 1.87·29-s − 0.169·31-s + 0.176·32-s − 1.61·35-s + 0.408·37-s + 0.0411·38-s − 0.660·40-s + 0.417·41-s + 0.197·43-s + 0.682·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.705486230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.705486230\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 4.17T + 5T^{2} \) |
| 7 | \( 1 - 2.28T + 7T^{2} \) |
| 11 | \( 1 - 4.52T + 11T^{2} \) |
| 13 | \( 1 - 2.72T + 13T^{2} \) |
| 19 | \( 1 - 0.253T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 0.944T + 31T^{2} \) |
| 37 | \( 1 - 2.48T + 37T^{2} \) |
| 41 | \( 1 - 2.67T + 41T^{2} \) |
| 43 | \( 1 - 1.29T + 43T^{2} \) |
| 47 | \( 1 - 3.55T + 47T^{2} \) |
| 53 | \( 1 + 0.457T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 + 1.92T + 61T^{2} \) |
| 67 | \( 1 - 3.71T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 - 7.82T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 3.11T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.043355873913551019196853248094, −7.48877846677768065288107176937, −6.87161800111923266791193944901, −6.02228432394590914020393902747, −5.11852224267690287727254206488, −4.21814187438528280037683492986, −3.92949038777217685471464043690, −3.27092110842346064409774710101, −1.89258720504176388048082774142, −0.835071374319511885634739358228,
0.835071374319511885634739358228, 1.89258720504176388048082774142, 3.27092110842346064409774710101, 3.92949038777217685471464043690, 4.21814187438528280037683492986, 5.11852224267690287727254206488, 6.02228432394590914020393902747, 6.87161800111923266791193944901, 7.48877846677768065288107176937, 8.043355873913551019196853248094
