L(s) = 1 | − 1.17·2-s + 3-s − 0.617·4-s + 3.63·5-s − 1.17·6-s − 0.818·7-s + 3.07·8-s + 9-s − 4.27·10-s + 11-s − 0.617·12-s + 3.69·13-s + 0.962·14-s + 3.63·15-s − 2.38·16-s + 2.16·17-s − 1.17·18-s + 6.52·19-s − 2.24·20-s − 0.818·21-s − 1.17·22-s − 2.86·23-s + 3.07·24-s + 8.24·25-s − 4.34·26-s + 27-s + 0.505·28-s + ⋯ |
L(s) = 1 | − 0.831·2-s + 0.577·3-s − 0.308·4-s + 1.62·5-s − 0.480·6-s − 0.309·7-s + 1.08·8-s + 0.333·9-s − 1.35·10-s + 0.301·11-s − 0.178·12-s + 1.02·13-s + 0.257·14-s + 0.939·15-s − 0.596·16-s + 0.524·17-s − 0.277·18-s + 1.49·19-s − 0.502·20-s − 0.178·21-s − 0.250·22-s − 0.597·23-s + 0.628·24-s + 1.64·25-s − 0.851·26-s + 0.192·27-s + 0.0954·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.874205542\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.874205542\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 + 1.17T + 2T^{2} \) |
| 5 | \( 1 - 3.63T + 5T^{2} \) |
| 7 | \( 1 + 0.818T + 7T^{2} \) |
| 13 | \( 1 - 3.69T + 13T^{2} \) |
| 17 | \( 1 - 2.16T + 17T^{2} \) |
| 19 | \( 1 - 6.52T + 19T^{2} \) |
| 23 | \( 1 + 2.86T + 23T^{2} \) |
| 29 | \( 1 + 8.99T + 29T^{2} \) |
| 31 | \( 1 - 4.22T + 31T^{2} \) |
| 37 | \( 1 + 9.05T + 37T^{2} \) |
| 41 | \( 1 - 5.62T + 41T^{2} \) |
| 43 | \( 1 - 4.76T + 43T^{2} \) |
| 47 | \( 1 - 6.54T + 47T^{2} \) |
| 53 | \( 1 - 2.05T + 53T^{2} \) |
| 59 | \( 1 - 3.99T + 59T^{2} \) |
| 67 | \( 1 + 8.12T + 67T^{2} \) |
| 71 | \( 1 + 6.74T + 71T^{2} \) |
| 73 | \( 1 - 4.15T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 1.52T + 83T^{2} \) |
| 89 | \( 1 + 2.32T + 89T^{2} \) |
| 97 | \( 1 - 3.41T + 97T^{2} \) |
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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
−9.226491321998185264949937181518, −8.683563116953790299845647886944, −7.74637019824493091215477181616, −6.99113968888195216058826015910, −5.89858078102665223040835796529, −5.39531697830249491874889020366, −4.11637198788268209419465976885, −3.13701603949687565542589070108, −1.88035248852768013926615104689, −1.13350089469919359270514534887,
1.13350089469919359270514534887, 1.88035248852768013926615104689, 3.13701603949687565542589070108, 4.11637198788268209419465976885, 5.39531697830249491874889020366, 5.89858078102665223040835796529, 6.99113968888195216058826015910, 7.74637019824493091215477181616, 8.683563116953790299845647886944, 9.226491321998185264949937181518