L(s) = 1 | + 3-s + 5-s + 4.86·7-s + 9-s + 0.902·11-s − 0.581·13-s + 15-s + 4.01·17-s − 1.30·19-s + 4.86·21-s + 2.61·23-s + 25-s + 27-s + 4.25·29-s + 4.64·31-s + 0.902·33-s + 4.86·35-s − 6.64·37-s − 0.581·39-s − 10.9·41-s − 2.99·43-s + 45-s − 2.18·47-s + 16.7·49-s + 4.01·51-s + 2.97·53-s + 0.902·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.84·7-s + 0.333·9-s + 0.272·11-s − 0.161·13-s + 0.258·15-s + 0.972·17-s − 0.299·19-s + 1.06·21-s + 0.545·23-s + 0.200·25-s + 0.192·27-s + 0.790·29-s + 0.834·31-s + 0.157·33-s + 0.823·35-s − 1.09·37-s − 0.0931·39-s − 1.70·41-s − 0.456·43-s + 0.149·45-s − 0.318·47-s + 2.38·49-s + 0.561·51-s + 0.408·53-s + 0.121·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.583323766\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.583323766\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + T \) |
good | 7 | \( 1 - 4.86T + 7T^{2} \) |
| 11 | \( 1 - 0.902T + 11T^{2} \) |
| 13 | \( 1 + 0.581T + 13T^{2} \) |
| 17 | \( 1 - 4.01T + 17T^{2} \) |
| 19 | \( 1 + 1.30T + 19T^{2} \) |
| 23 | \( 1 - 2.61T + 23T^{2} \) |
| 29 | \( 1 - 4.25T + 29T^{2} \) |
| 31 | \( 1 - 4.64T + 31T^{2} \) |
| 37 | \( 1 + 6.64T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 2.99T + 43T^{2} \) |
| 47 | \( 1 + 2.18T + 47T^{2} \) |
| 53 | \( 1 - 2.97T + 53T^{2} \) |
| 59 | \( 1 - 0.793T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 71 | \( 1 - 3.53T + 71T^{2} \) |
| 73 | \( 1 - 6.74T + 73T^{2} \) |
| 79 | \( 1 - 5.46T + 79T^{2} \) |
| 83 | \( 1 + 0.259T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 9.07T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.394462958450466725113905962093, −7.899384992759982368037387289537, −7.11557844005457545302692722544, −6.29321982264806950186138488121, −5.12947392951274215939069742311, −4.90515610096546725335458409921, −3.84899554911394652193383843712, −2.85961979279554001326932442421, −1.85669906538231883078903290778, −1.20829014664377172560016926980,
1.20829014664377172560016926980, 1.85669906538231883078903290778, 2.85961979279554001326932442421, 3.84899554911394652193383843712, 4.90515610096546725335458409921, 5.12947392951274215939069742311, 6.29321982264806950186138488121, 7.11557844005457545302692722544, 7.899384992759982368037387289537, 8.394462958450466725113905962093