L(s) = 1 | − 2.74·2-s − 0.0877·3-s + 5.52·4-s + 0.240·6-s − 1.38·7-s − 9.67·8-s − 2.99·9-s − 1.68·11-s − 0.484·12-s − 1.34·13-s + 3.80·14-s + 15.4·16-s − 3.06·17-s + 8.20·18-s + 8.07·19-s + 0.121·21-s + 4.62·22-s − 2.18·23-s + 0.848·24-s + 3.69·26-s + 0.525·27-s − 7.67·28-s − 8.72·29-s + 7.09·31-s − 23.1·32-s + 0.147·33-s + 8.41·34-s + ⋯ |
L(s) = 1 | − 1.94·2-s − 0.0506·3-s + 2.76·4-s + 0.0982·6-s − 0.524·7-s − 3.42·8-s − 0.997·9-s − 0.508·11-s − 0.139·12-s − 0.373·13-s + 1.01·14-s + 3.87·16-s − 0.743·17-s + 1.93·18-s + 1.85·19-s + 0.0265·21-s + 0.986·22-s − 0.455·23-s + 0.173·24-s + 0.723·26-s + 0.101·27-s − 1.45·28-s − 1.61·29-s + 1.27·31-s − 4.09·32-s + 0.0257·33-s + 1.44·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 3 | \( 1 + 0.0877T + 3T^{2} \) |
| 7 | \( 1 + 1.38T + 7T^{2} \) |
| 11 | \( 1 + 1.68T + 11T^{2} \) |
| 13 | \( 1 + 1.34T + 13T^{2} \) |
| 17 | \( 1 + 3.06T + 17T^{2} \) |
| 19 | \( 1 - 8.07T + 19T^{2} \) |
| 23 | \( 1 + 2.18T + 23T^{2} \) |
| 29 | \( 1 + 8.72T + 29T^{2} \) |
| 31 | \( 1 - 7.09T + 31T^{2} \) |
| 37 | \( 1 + 0.976T + 37T^{2} \) |
| 41 | \( 1 - 3.27T + 41T^{2} \) |
| 43 | \( 1 - 8.18T + 43T^{2} \) |
| 47 | \( 1 - 8.16T + 47T^{2} \) |
| 53 | \( 1 + 2.61T + 53T^{2} \) |
| 59 | \( 1 - 2.45T + 59T^{2} \) |
| 61 | \( 1 + 3.69T + 61T^{2} \) |
| 67 | \( 1 - 5.99T + 67T^{2} \) |
| 71 | \( 1 - 9.10T + 71T^{2} \) |
| 73 | \( 1 - 4.03T + 73T^{2} \) |
| 79 | \( 1 + 0.0889T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 0.520T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−7.74912395030222428497892361434, −7.40634746561228747863418224880, −6.51934067721314781007041438221, −5.89808291180849272170216737757, −5.19861404936030358294082164449, −3.61580087332328187887415459089, −2.76657686624504532792630096275, −2.22655245869899463339282198360, −0.933194433359786461072094256390, 0,
0.933194433359786461072094256390, 2.22655245869899463339282198360, 2.76657686624504532792630096275, 3.61580087332328187887415459089, 5.19861404936030358294082164449, 5.89808291180849272170216737757, 6.51934067721314781007041438221, 7.40634746561228747863418224880, 7.74912395030222428497892361434