L(s) = 1 | − 1.03·5-s + 3.86·7-s + 1.46·11-s − 13-s − 5.65·17-s − 1.79·19-s − 6.92·23-s − 3.92·25-s − 2.07·29-s + 3.86·31-s − 3.99·35-s − 2·37-s + 1.03·41-s − 5.65·43-s − 2.53·47-s + 7.92·49-s − 5.65·53-s − 1.51·55-s − 9.46·59-s − 8.92·61-s + 1.03·65-s + 1.79·67-s − 2.53·71-s + 11.8·73-s + 5.65·77-s − 6.53·83-s + 5.85·85-s + ⋯ |
L(s) = 1 | − 0.462·5-s + 1.46·7-s + 0.441·11-s − 0.277·13-s − 1.37·17-s − 0.411·19-s − 1.44·23-s − 0.785·25-s − 0.384·29-s + 0.693·31-s − 0.676·35-s − 0.328·37-s + 0.161·41-s − 0.862·43-s − 0.369·47-s + 1.13·49-s − 0.777·53-s − 0.204·55-s − 1.23·59-s − 1.14·61-s + 0.128·65-s + 0.219·67-s − 0.300·71-s + 1.38·73-s + 0.644·77-s − 0.717·83-s + 0.635·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + 1.03T + 5T^{2} \) |
| 7 | \( 1 - 3.86T + 7T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 + 1.79T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + 2.07T + 29T^{2} \) |
| 31 | \( 1 - 3.86T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 1.03T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 + 2.53T + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 - 1.79T + 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 6.53T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.048246131869395849780438206379, −7.66272791795701991947576963077, −6.65131894232556576836704528750, −5.97540434550894653590090730319, −4.86100091456084649821553170486, −4.43621940317334507971570127482, −3.63236574380473833266677102175, −2.26364901843645638529988389610, −1.60899409836233655369460576161, 0,
1.60899409836233655369460576161, 2.26364901843645638529988389610, 3.63236574380473833266677102175, 4.43621940317334507971570127482, 4.86100091456084649821553170486, 5.97540434550894653590090730319, 6.65131894232556576836704528750, 7.66272791795701991947576963077, 8.048246131869395849780438206379