| L(s) = 1 | + 2.95·5-s − 3.03·7-s − 4.44·17-s + 3.54·19-s − 8.07·23-s + 3.73·25-s − 6.58·29-s − 3.23·31-s − 8.98·35-s − 4.77·37-s + 7.03·41-s + 10.7·43-s + 12.0·47-s + 2.23·49-s + 12.7·53-s + 2.79·59-s − 5.81·61-s + 14.6·67-s + 15.3·71-s − 4.70·73-s + 7.33·79-s + 7.65·83-s − 13.1·85-s − 0.858·89-s + 10.4·95-s + 2.67·97-s + 10.2·101-s + ⋯ |
| L(s) = 1 | + 1.32·5-s − 1.14·7-s − 1.07·17-s + 0.812·19-s − 1.68·23-s + 0.747·25-s − 1.22·29-s − 0.581·31-s − 1.51·35-s − 0.784·37-s + 1.09·41-s + 1.63·43-s + 1.75·47-s + 0.319·49-s + 1.74·53-s + 0.363·59-s − 0.745·61-s + 1.78·67-s + 1.82·71-s − 0.551·73-s + 0.825·79-s + 0.839·83-s − 1.42·85-s − 0.0909·89-s + 1.07·95-s + 0.271·97-s + 1.01·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.954014040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.954014040\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 5 | \( 1 - 2.95T + 5T^{2} \) |
| 7 | \( 1 + 3.03T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 4.44T + 17T^{2} \) |
| 19 | \( 1 - 3.54T + 19T^{2} \) |
| 23 | \( 1 + 8.07T + 23T^{2} \) |
| 29 | \( 1 + 6.58T + 29T^{2} \) |
| 31 | \( 1 + 3.23T + 31T^{2} \) |
| 37 | \( 1 + 4.77T + 37T^{2} \) |
| 41 | \( 1 - 7.03T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 2.79T + 59T^{2} \) |
| 61 | \( 1 + 5.81T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 + 4.70T + 73T^{2} \) |
| 79 | \( 1 - 7.33T + 79T^{2} \) |
| 83 | \( 1 - 7.65T + 83T^{2} \) |
| 89 | \( 1 + 0.858T + 89T^{2} \) |
| 97 | \( 1 - 2.67T + 97T^{2} \) |
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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
−7.68059114294879747613324163410, −7.06368505216707111547492974692, −6.30334962091276610277518642401, −5.81769088120040888793609231967, −5.33477982218446847332327253298, −4.13431408491405876503071489395, −3.56497139884231080681806086456, −2.38021236025798101988904015014, −2.07992929507538310943784740604, −0.66400881673180617645079980608,
0.66400881673180617645079980608, 2.07992929507538310943784740604, 2.38021236025798101988904015014, 3.56497139884231080681806086456, 4.13431408491405876503071489395, 5.33477982218446847332327253298, 5.81769088120040888793609231967, 6.30334962091276610277518642401, 7.06368505216707111547492974692, 7.68059114294879747613324163410
