L(s) = 1 | − 5-s + 2·7-s + 4.44·11-s − 13-s − 6.89·17-s + 0.449·19-s − 6.44·23-s + 25-s − 4·29-s − 4.44·31-s − 2·35-s + 4.89·37-s − 10.8·41-s + 11.3·43-s + 2·47-s − 3·49-s − 1.10·53-s − 4.44·55-s − 9.34·59-s − 5.79·61-s + 65-s − 5.10·67-s + 3.55·71-s − 14.6·73-s + 8.89·77-s + 4.89·79-s − 2·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.755·7-s + 1.34·11-s − 0.277·13-s − 1.67·17-s + 0.103·19-s − 1.34·23-s + 0.200·25-s − 0.742·29-s − 0.799·31-s − 0.338·35-s + 0.805·37-s − 1.70·41-s + 1.73·43-s + 0.291·47-s − 0.428·49-s − 0.151·53-s − 0.599·55-s − 1.21·59-s − 0.742·61-s + 0.124·65-s − 0.623·67-s + 0.421·71-s − 1.72·73-s + 1.01·77-s + 0.551·79-s − 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 4.44T + 11T^{2} \) |
| 17 | \( 1 + 6.89T + 17T^{2} \) |
| 19 | \( 1 - 0.449T + 19T^{2} \) |
| 23 | \( 1 + 6.44T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 4.44T + 31T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 1.10T + 53T^{2} \) |
| 59 | \( 1 + 9.34T + 59T^{2} \) |
| 61 | \( 1 + 5.79T + 61T^{2} \) |
| 67 | \( 1 + 5.10T + 67T^{2} \) |
| 71 | \( 1 - 3.55T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 - 4.89T + 79T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 7.79T + 97T^{2} \) |
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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
−7.84751013594059444218594388666, −7.33469462211241109564909565564, −6.47573440382732111789257703802, −5.87746010886331827597204936922, −4.72854018272802603278693727786, −4.26837272076479963526390853975, −3.51915981516653765988320901798, −2.24704576644844845324494842483, −1.49108367490569580390339329579, 0,
1.49108367490569580390339329579, 2.24704576644844845324494842483, 3.51915981516653765988320901798, 4.26837272076479963526390853975, 4.72854018272802603278693727786, 5.87746010886331827597204936922, 6.47573440382732111789257703802, 7.33469462211241109564909565564, 7.84751013594059444218594388666