L(s) = 1 | − 0.935·3-s + 3.41·5-s − 3.69·7-s − 2.12·9-s + 1.09·11-s + 0.0975·13-s − 3.19·15-s − 4.03·17-s − 5.04·19-s + 3.45·21-s + 0.592·23-s + 6.66·25-s + 4.79·27-s − 3.45·29-s + 0.372·31-s − 1.02·33-s − 12.6·35-s + 6.18·37-s − 0.0912·39-s + 11.3·41-s + 5.37·43-s − 7.25·45-s + 10.3·47-s + 6.64·49-s + 3.77·51-s − 11.2·53-s + 3.74·55-s + ⋯ |
L(s) = 1 | − 0.540·3-s + 1.52·5-s − 1.39·7-s − 0.708·9-s + 0.330·11-s + 0.0270·13-s − 0.825·15-s − 0.978·17-s − 1.15·19-s + 0.753·21-s + 0.123·23-s + 1.33·25-s + 0.922·27-s − 0.642·29-s + 0.0669·31-s − 0.178·33-s − 2.13·35-s + 1.01·37-s − 0.0146·39-s + 1.77·41-s + 0.819·43-s − 1.08·45-s + 1.51·47-s + 0.948·49-s + 0.528·51-s − 1.54·53-s + 0.504·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.368911215\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.368911215\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 + 0.935T + 3T^{2} \) |
| 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 - 1.09T + 11T^{2} \) |
| 13 | \( 1 - 0.0975T + 13T^{2} \) |
| 17 | \( 1 + 4.03T + 17T^{2} \) |
| 19 | \( 1 + 5.04T + 19T^{2} \) |
| 23 | \( 1 - 0.592T + 23T^{2} \) |
| 29 | \( 1 + 3.45T + 29T^{2} \) |
| 31 | \( 1 - 0.372T + 31T^{2} \) |
| 37 | \( 1 - 6.18T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 5.37T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 7.25T + 59T^{2} \) |
| 61 | \( 1 - 0.601T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 0.624T + 71T^{2} \) |
| 73 | \( 1 - 9.71T + 73T^{2} \) |
| 79 | \( 1 - 6.82T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 7.70T + 97T^{2} \) |
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
−8.827484338911046433603675039270, −7.58214963144294325255438852522, −6.49021193053530532258748362219, −6.25159133223709034554872527504, −5.81183003346513322737827973902, −4.83193662148258487150303275024, −3.84827390047568204310396789418, −2.69994172503049269846120679052, −2.16959431093101979486017406370, −0.65882802352103436358887577836,
0.65882802352103436358887577836, 2.16959431093101979486017406370, 2.69994172503049269846120679052, 3.84827390047568204310396789418, 4.83193662148258487150303275024, 5.81183003346513322737827973902, 6.25159133223709034554872527504, 6.49021193053530532258748362219, 7.58214963144294325255438852522, 8.827484338911046433603675039270