| L(s) = 1 | − 1.36·2-s − 0.399·3-s − 0.135·4-s + 0.546·6-s − 3.64·7-s + 2.91·8-s − 2.84·9-s − 3.27·11-s + 0.0541·12-s + 4.97·14-s − 3.71·16-s − 2.17·17-s + 3.87·18-s + 0.857·19-s + 1.45·21-s + 4.47·22-s − 0.765·23-s − 1.16·24-s + 2.33·27-s + 0.493·28-s − 3.06·29-s − 8.41·31-s − 0.764·32-s + 1.31·33-s + 2.97·34-s + 0.384·36-s − 10.8·37-s + ⋯ |
| L(s) = 1 | − 0.965·2-s − 0.230·3-s − 0.0676·4-s + 0.222·6-s − 1.37·7-s + 1.03·8-s − 0.946·9-s − 0.988·11-s + 0.0156·12-s + 1.32·14-s − 0.927·16-s − 0.527·17-s + 0.914·18-s + 0.196·19-s + 0.317·21-s + 0.954·22-s − 0.159·23-s − 0.238·24-s + 0.449·27-s + 0.0932·28-s − 0.569·29-s − 1.51·31-s − 0.135·32-s + 0.228·33-s + 0.509·34-s + 0.0640·36-s − 1.78·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02446905758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02446905758\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.36T + 2T^{2} \) |
| 3 | \( 1 + 0.399T + 3T^{2} \) |
| 7 | \( 1 + 3.64T + 7T^{2} \) |
| 11 | \( 1 + 3.27T + 11T^{2} \) |
| 17 | \( 1 + 2.17T + 17T^{2} \) |
| 19 | \( 1 - 0.857T + 19T^{2} \) |
| 23 | \( 1 + 0.765T + 23T^{2} \) |
| 29 | \( 1 + 3.06T + 29T^{2} \) |
| 31 | \( 1 + 8.41T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 7.61T + 41T^{2} \) |
| 43 | \( 1 - 3.83T + 43T^{2} \) |
| 47 | \( 1 - 4.68T + 47T^{2} \) |
| 53 | \( 1 + 3.04T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 4.64T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 7.68T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 1.18T + 83T^{2} \) |
| 89 | \( 1 + 3.26T + 89T^{2} \) |
| 97 | \( 1 - 5.25T + 97T^{2} \) |
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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
−8.731761985719585235311271955991, −7.65557277316252464503152835533, −7.19435776124006928743860987254, −6.23755922569991062511580761874, −5.55062401610756423793133876640, −4.77682606427974357085207290805, −3.62491617393724756116836930045, −2.90110627496961215604143780962, −1.77280066767922757678450406212, −0.099128825676090319485215808352,
0.099128825676090319485215808352, 1.77280066767922757678450406212, 2.90110627496961215604143780962, 3.62491617393724756116836930045, 4.77682606427974357085207290805, 5.55062401610756423793133876640, 6.23755922569991062511580761874, 7.19435776124006928743860987254, 7.65557277316252464503152835533, 8.731761985719585235311271955991
