| L(s) = 1 | − 2.41·2-s − 3-s + 3.82·4-s + 2.41·6-s − 0.414·7-s − 4.41·8-s + 9-s − 11-s − 3.82·12-s + 2.82·13-s + 0.999·14-s + 2.99·16-s − 2.41·17-s − 2.41·18-s + 6.41·19-s + 0.414·21-s + 2.41·22-s − 23-s + 4.41·24-s − 6.82·26-s − 27-s − 1.58·28-s + 1.17·29-s − 8.48·31-s + 1.58·32-s + 33-s + 5.82·34-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.577·3-s + 1.91·4-s + 0.985·6-s − 0.156·7-s − 1.56·8-s + 0.333·9-s − 0.301·11-s − 1.10·12-s + 0.784·13-s + 0.267·14-s + 0.749·16-s − 0.585·17-s − 0.569·18-s + 1.47·19-s + 0.0903·21-s + 0.514·22-s − 0.208·23-s + 0.901·24-s − 1.33·26-s − 0.192·27-s − 0.299·28-s + 0.217·29-s − 1.52·31-s + 0.280·32-s + 0.174·33-s + 0.999·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5108467140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5108467140\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 7 | \( 1 + 0.414T + 7T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 2.41T + 17T^{2} \) |
| 19 | \( 1 - 6.41T + 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 - 1.17T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 - 0.171T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 7.48T + 47T^{2} \) |
| 53 | \( 1 + 7.65T + 53T^{2} \) |
| 59 | \( 1 - 11T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 - 0.343T + 67T^{2} \) |
| 71 | \( 1 - 7.82T + 71T^{2} \) |
| 73 | \( 1 - 8.82T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 4.48T + 83T^{2} \) |
| 89 | \( 1 - 3.65T + 89T^{2} \) |
| 97 | \( 1 + 5.82T + 97T^{2} \) |
| 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
−10.09766746946000514003095168972, −9.411609650044072762521810834062, −8.663637355351682074465559656083, −7.75036923751843003731406402542, −7.04181707120955560905574405964, −6.17566666897077187845548492209, −5.14076392050016555645541912914, −3.55880570075872032781970268036, −2.06395509747984882662455571235, −0.77242910565716435963065767883,
0.77242910565716435963065767883, 2.06395509747984882662455571235, 3.55880570075872032781970268036, 5.14076392050016555645541912914, 6.17566666897077187845548492209, 7.04181707120955560905574405964, 7.75036923751843003731406402542, 8.663637355351682074465559656083, 9.411609650044072762521810834062, 10.09766746946000514003095168972
