| L(s) = 1 | + 2.81·2-s + 5.94·4-s + 0.621·5-s + 3.29·7-s + 11.1·8-s + 1.75·10-s − 11-s − 1.50·13-s + 9.27·14-s + 19.4·16-s + 6.84·17-s − 4.68·19-s + 3.69·20-s − 2.81·22-s − 6.01·23-s − 4.61·25-s − 4.25·26-s + 19.5·28-s + 0.884·29-s − 0.545·31-s + 32.5·32-s + 19.2·34-s + 2.04·35-s + 6.78·37-s − 13.2·38-s + 6.90·40-s − 1.45·41-s + ⋯ |
| L(s) = 1 | + 1.99·2-s + 2.97·4-s + 0.277·5-s + 1.24·7-s + 3.92·8-s + 0.553·10-s − 0.301·11-s − 0.418·13-s + 2.48·14-s + 4.85·16-s + 1.65·17-s − 1.07·19-s + 0.825·20-s − 0.600·22-s − 1.25·23-s − 0.922·25-s − 0.834·26-s + 3.69·28-s + 0.164·29-s − 0.0979·31-s + 5.74·32-s + 3.30·34-s + 0.345·35-s + 1.11·37-s − 2.14·38-s + 1.09·40-s − 0.227·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.957428934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.957428934\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.81T + 2T^{2} \) |
| 5 | \( 1 - 0.621T + 5T^{2} \) |
| 7 | \( 1 - 3.29T + 7T^{2} \) |
| 13 | \( 1 + 1.50T + 13T^{2} \) |
| 17 | \( 1 - 6.84T + 17T^{2} \) |
| 19 | \( 1 + 4.68T + 19T^{2} \) |
| 23 | \( 1 + 6.01T + 23T^{2} \) |
| 29 | \( 1 - 0.884T + 29T^{2} \) |
| 31 | \( 1 + 0.545T + 31T^{2} \) |
| 37 | \( 1 - 6.78T + 37T^{2} \) |
| 41 | \( 1 + 1.45T + 41T^{2} \) |
| 43 | \( 1 - 1.14T + 43T^{2} \) |
| 47 | \( 1 + 9.26T + 47T^{2} \) |
| 53 | \( 1 + 3.50T + 53T^{2} \) |
| 59 | \( 1 - 9.56T + 59T^{2} \) |
| 67 | \( 1 + 9.11T + 67T^{2} \) |
| 71 | \( 1 + 6.11T + 71T^{2} \) |
| 73 | \( 1 - 2.67T + 73T^{2} \) |
| 79 | \( 1 + 5.91T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 4.13T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−7.894062931997380245892809785000, −7.24551163952679555246464378771, −6.26210829352077645916247878009, −5.75960580417428761728225986229, −5.15707515505865387993011430649, −4.47164627314841909033410510455, −3.88948766601923010871652808047, −2.93277920491375262500751210477, −2.09867093196655415369025244959, −1.47033213322724801590760267578,
1.47033213322724801590760267578, 2.09867093196655415369025244959, 2.93277920491375262500751210477, 3.88948766601923010871652808047, 4.47164627314841909033410510455, 5.15707515505865387993011430649, 5.75960580417428761728225986229, 6.26210829352077645916247878009, 7.24551163952679555246464378771, 7.894062931997380245892809785000
