| L(s) = 1 | − 0.766·2-s − 3-s − 1.41·4-s − 0.561·5-s + 0.766·6-s + 2.61·8-s + 9-s + 0.430·10-s − 3.11·11-s + 1.41·12-s − 5.92·13-s + 0.561·15-s + 0.823·16-s − 3.15·17-s − 0.766·18-s + 0.142·19-s + 0.793·20-s + 2.38·22-s − 4.62·23-s − 2.61·24-s − 4.68·25-s + 4.53·26-s − 27-s − 8.30·29-s − 0.430·30-s + 0.336·31-s − 5.86·32-s + ⋯ |
| L(s) = 1 | − 0.541·2-s − 0.577·3-s − 0.706·4-s − 0.251·5-s + 0.312·6-s + 0.924·8-s + 0.333·9-s + 0.136·10-s − 0.938·11-s + 0.407·12-s − 1.64·13-s + 0.145·15-s + 0.205·16-s − 0.764·17-s − 0.180·18-s + 0.0326·19-s + 0.177·20-s + 0.508·22-s − 0.965·23-s − 0.533·24-s − 0.936·25-s + 0.890·26-s − 0.192·27-s − 1.54·29-s − 0.0785·30-s + 0.0604·31-s − 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03388972134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03388972134\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 0.766T + 2T^{2} \) |
| 5 | \( 1 + 0.561T + 5T^{2} \) |
| 11 | \( 1 + 3.11T + 11T^{2} \) |
| 13 | \( 1 + 5.92T + 13T^{2} \) |
| 17 | \( 1 + 3.15T + 17T^{2} \) |
| 19 | \( 1 - 0.142T + 19T^{2} \) |
| 23 | \( 1 + 4.62T + 23T^{2} \) |
| 29 | \( 1 + 8.30T + 29T^{2} \) |
| 31 | \( 1 - 0.336T + 31T^{2} \) |
| 37 | \( 1 + 9.44T + 37T^{2} \) |
| 43 | \( 1 - 9.28T + 43T^{2} \) |
| 47 | \( 1 - 7.35T + 47T^{2} \) |
| 53 | \( 1 + 8.42T + 53T^{2} \) |
| 59 | \( 1 + 0.911T + 59T^{2} \) |
| 61 | \( 1 - 3.48T + 61T^{2} \) |
| 67 | \( 1 + 0.537T + 67T^{2} \) |
| 71 | \( 1 + 0.835T + 71T^{2} \) |
| 73 | \( 1 + 8.54T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 5.99T + 83T^{2} \) |
| 89 | \( 1 + 18.7T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
| 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
−7.88461052993247245771745656253, −7.57817393626928215549255791016, −6.90374338255163067034799590394, −5.68417032140352001546442006834, −5.31100555128031564760462723427, −4.43660063092103803050951741212, −3.92893633271572685963253605102, −2.59514457564335695390994932051, −1.71364668030420616033672230225, −0.10529784813970482595904935063,
0.10529784813970482595904935063, 1.71364668030420616033672230225, 2.59514457564335695390994932051, 3.92893633271572685963253605102, 4.43660063092103803050951741212, 5.31100555128031564760462723427, 5.68417032140352001546442006834, 6.90374338255163067034799590394, 7.57817393626928215549255791016, 7.88461052993247245771745656253
