| L(s) = 1 | − 0.496·2-s − 3-s − 1.75·4-s + 5-s + 0.496·6-s − 1.84·7-s + 1.86·8-s + 9-s − 0.496·10-s − 2.52·11-s + 1.75·12-s − 13-s + 0.915·14-s − 15-s + 2.58·16-s − 4.39·17-s − 0.496·18-s + 2.58·19-s − 1.75·20-s + 1.84·21-s + 1.25·22-s − 6.20·23-s − 1.86·24-s + 25-s + 0.496·26-s − 27-s + 3.23·28-s + ⋯ |
| L(s) = 1 | − 0.350·2-s − 0.577·3-s − 0.876·4-s + 0.447·5-s + 0.202·6-s − 0.697·7-s + 0.658·8-s + 0.333·9-s − 0.156·10-s − 0.762·11-s + 0.506·12-s − 0.277·13-s + 0.244·14-s − 0.258·15-s + 0.646·16-s − 1.06·17-s − 0.116·18-s + 0.592·19-s − 0.392·20-s + 0.402·21-s + 0.267·22-s − 1.29·23-s − 0.380·24-s + 0.200·25-s + 0.0972·26-s − 0.192·27-s + 0.611·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4207560191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4207560191\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 0.496T + 2T^{2} \) |
| 7 | \( 1 + 1.84T + 7T^{2} \) |
| 11 | \( 1 + 2.52T + 11T^{2} \) |
| 17 | \( 1 + 4.39T + 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 23 | \( 1 + 6.20T + 23T^{2} \) |
| 29 | \( 1 + 6.14T + 29T^{2} \) |
| 37 | \( 1 - 4.25T + 37T^{2} \) |
| 41 | \( 1 + 7.02T + 41T^{2} \) |
| 43 | \( 1 - 7.37T + 43T^{2} \) |
| 47 | \( 1 - 4.20T + 47T^{2} \) |
| 53 | \( 1 - 5.38T + 53T^{2} \) |
| 59 | \( 1 + 0.991T + 59T^{2} \) |
| 61 | \( 1 + 4.39T + 61T^{2} \) |
| 67 | \( 1 + 9.05T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 1.45T + 73T^{2} \) |
| 79 | \( 1 + 7.77T + 79T^{2} \) |
| 83 | \( 1 + 1.73T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 - 9.82T + 97T^{2} \) |
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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
−8.039641264269074380912718248079, −7.45029193291539328709979476140, −6.63782969160888198953420114816, −5.79499873524164624209908328063, −5.35088711376564297453471466564, −4.47117122787695254753438199747, −3.81929598108622888433111268689, −2.70568989151484259964507902356, −1.69027093449774089179545692020, −0.36762323134117193005128420962,
0.36762323134117193005128420962, 1.69027093449774089179545692020, 2.70568989151484259964507902356, 3.81929598108622888433111268689, 4.47117122787695254753438199747, 5.35088711376564297453471466564, 5.79499873524164624209908328063, 6.63782969160888198953420114816, 7.45029193291539328709979476140, 8.039641264269074380912718248079
