| L(s) = 1 | + 2.21·2-s − 1.45·3-s + 2.89·4-s + 5-s − 3.21·6-s + 7-s + 1.97·8-s − 0.891·9-s + 2.21·10-s − 4.19·12-s − 6.19·13-s + 2.21·14-s − 1.45·15-s − 1.42·16-s + 6.75·17-s − 1.97·18-s − 6.55·19-s + 2.89·20-s − 1.45·21-s − 1.75·23-s − 2.86·24-s + 25-s − 13.7·26-s + 5.65·27-s + 2.89·28-s + 3.32·29-s − 3.21·30-s + ⋯ |
| L(s) = 1 | + 1.56·2-s − 0.838·3-s + 1.44·4-s + 0.447·5-s − 1.31·6-s + 0.377·7-s + 0.696·8-s − 0.297·9-s + 0.699·10-s − 1.21·12-s − 1.71·13-s + 0.591·14-s − 0.374·15-s − 0.355·16-s + 1.63·17-s − 0.464·18-s − 1.50·19-s + 0.646·20-s − 0.316·21-s − 0.366·23-s − 0.584·24-s + 0.200·25-s − 2.68·26-s + 1.08·27-s + 0.546·28-s + 0.617·29-s − 0.586·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.21T + 2T^{2} \) |
| 3 | \( 1 + 1.45T + 3T^{2} \) |
| 13 | \( 1 + 6.19T + 13T^{2} \) |
| 17 | \( 1 - 6.75T + 17T^{2} \) |
| 19 | \( 1 + 6.55T + 19T^{2} \) |
| 23 | \( 1 + 1.75T + 23T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 - 5.76T + 31T^{2} \) |
| 37 | \( 1 + 9.70T + 37T^{2} \) |
| 41 | \( 1 + 7.98T + 41T^{2} \) |
| 43 | \( 1 + 5.27T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 5.33T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 7.27T + 67T^{2} \) |
| 71 | \( 1 - 8.82T + 71T^{2} \) |
| 73 | \( 1 + 0.811T + 73T^{2} \) |
| 79 | \( 1 - 4.53T + 79T^{2} \) |
| 83 | \( 1 - 0.639T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 7.25T + 97T^{2} \) |
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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
−7.85528342905723096270140165286, −6.78310640796795240493887891885, −6.39612565591760217759196600212, −5.57291025766598781735184579456, −4.96549606366245252533201261184, −4.69481152837569601797396854601, −3.46109838602505941878583850629, −2.70149793061920737179052969462, −1.75752198432567331360591350244, 0,
1.75752198432567331360591350244, 2.70149793061920737179052969462, 3.46109838602505941878583850629, 4.69481152837569601797396854601, 4.96549606366245252533201261184, 5.57291025766598781735184579456, 6.39612565591760217759196600212, 6.78310640796795240493887891885, 7.85528342905723096270140165286
