| L(s) = 1 | + 2.28·2-s + 3.22·4-s + 2.86·5-s + 3.60·7-s + 2.79·8-s + 6.53·10-s + 11-s − 3.97·13-s + 8.22·14-s − 0.0663·16-s + 2.89·17-s + 3.91·19-s + 9.21·20-s + 2.28·22-s − 4.02·23-s + 3.18·25-s − 9.07·26-s + 11.5·28-s + 3.89·29-s − 0.470·31-s − 5.73·32-s + 6.62·34-s + 10.2·35-s − 1.32·37-s + 8.94·38-s + 7.98·40-s − 2.18·41-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 1.61·4-s + 1.27·5-s + 1.36·7-s + 0.986·8-s + 2.06·10-s + 0.301·11-s − 1.10·13-s + 2.19·14-s − 0.0165·16-s + 0.702·17-s + 0.897·19-s + 2.06·20-s + 0.487·22-s − 0.839·23-s + 0.636·25-s − 1.78·26-s + 2.19·28-s + 0.722·29-s − 0.0845·31-s − 1.01·32-s + 1.13·34-s + 1.74·35-s − 0.217·37-s + 1.45·38-s + 1.26·40-s − 0.340·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.970211056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.970211056\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 2.28T + 2T^{2} \) |
| 5 | \( 1 - 2.86T + 5T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 13 | \( 1 + 3.97T + 13T^{2} \) |
| 17 | \( 1 - 2.89T + 17T^{2} \) |
| 19 | \( 1 - 3.91T + 19T^{2} \) |
| 23 | \( 1 + 4.02T + 23T^{2} \) |
| 29 | \( 1 - 3.89T + 29T^{2} \) |
| 31 | \( 1 + 0.470T + 31T^{2} \) |
| 37 | \( 1 + 1.32T + 37T^{2} \) |
| 41 | \( 1 + 2.18T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 6.67T + 47T^{2} \) |
| 53 | \( 1 + 2.86T + 53T^{2} \) |
| 59 | \( 1 + 6.78T + 59T^{2} \) |
| 67 | \( 1 - 6.63T + 67T^{2} \) |
| 71 | \( 1 + 2.50T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 + 1.93T + 79T^{2} \) |
| 83 | \( 1 - 9.23T + 83T^{2} \) |
| 89 | \( 1 - 6.85T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−7.70360293838676431500983944204, −7.29649123181701852555099926322, −6.22985667147478138659215681175, −5.77835076123644972813993052045, −5.09591605890014154214586774087, −4.70223321208285992192225531023, −3.81977088903762630865032064792, −2.76435851338333618915255554460, −2.15291229093081796738705693817, −1.32428528288063266477629619528,
1.32428528288063266477629619528, 2.15291229093081796738705693817, 2.76435851338333618915255554460, 3.81977088903762630865032064792, 4.70223321208285992192225531023, 5.09591605890014154214586774087, 5.77835076123644972813993052045, 6.22985667147478138659215681175, 7.29649123181701852555099926322, 7.70360293838676431500983944204
