L(s) = 1 | + 1.21·2-s + 1.31·3-s − 0.525·4-s + 5-s + 1.59·6-s − 3.06·8-s − 1.28·9-s + 1.21·10-s + 11-s − 0.688·12-s + 2.68·13-s + 1.31·15-s − 2.67·16-s + 4.68·17-s − 1.55·18-s + 8.23·19-s − 0.525·20-s + 1.21·22-s − 4.14·23-s − 4.02·24-s + 25-s + 3.26·26-s − 5.61·27-s + 5.05·29-s + 1.59·30-s − 5.39·31-s + 2.88·32-s + ⋯ |
L(s) = 1 | + 0.858·2-s + 0.756·3-s − 0.262·4-s + 0.447·5-s + 0.649·6-s − 1.08·8-s − 0.426·9-s + 0.384·10-s + 0.301·11-s − 0.198·12-s + 0.745·13-s + 0.338·15-s − 0.668·16-s + 1.13·17-s − 0.366·18-s + 1.88·19-s − 0.117·20-s + 0.258·22-s − 0.864·23-s − 0.820·24-s + 0.200·25-s + 0.640·26-s − 1.08·27-s + 0.937·29-s + 0.290·30-s − 0.969·31-s + 0.510·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.545148767\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.545148767\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 3 | \( 1 - 1.31T + 3T^{2} \) |
| 13 | \( 1 - 2.68T + 13T^{2} \) |
| 17 | \( 1 - 4.68T + 17T^{2} \) |
| 19 | \( 1 - 8.23T + 19T^{2} \) |
| 23 | \( 1 + 4.14T + 23T^{2} \) |
| 29 | \( 1 - 5.05T + 29T^{2} \) |
| 31 | \( 1 + 5.39T + 31T^{2} \) |
| 37 | \( 1 - 4.76T + 37T^{2} \) |
| 41 | \( 1 - 1.16T + 41T^{2} \) |
| 43 | \( 1 - 5.95T + 43T^{2} \) |
| 47 | \( 1 - 8.68T + 47T^{2} \) |
| 53 | \( 1 + 0.769T + 53T^{2} \) |
| 59 | \( 1 + 4.02T + 59T^{2} \) |
| 61 | \( 1 - 0.407T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 2.19T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 8.51T + 79T^{2} \) |
| 83 | \( 1 - 9.90T + 83T^{2} \) |
| 89 | \( 1 - 9.80T + 89T^{2} \) |
| 97 | \( 1 + 8.85T + 97T^{2} \) |
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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
−9.022860087613765962016069408275, −8.067360394815936032060835518047, −7.46540798749994557457233480184, −6.13698867291239538994382008555, −5.75292630469813870874575019884, −4.96490222140628190201365359973, −3.83314358220441212368194207507, −3.33110215329385619342542164004, −2.50075049751627098615027948491, −1.05955041249915747520289383113,
1.05955041249915747520289383113, 2.50075049751627098615027948491, 3.33110215329385619342542164004, 3.83314358220441212368194207507, 4.96490222140628190201365359973, 5.75292630469813870874575019884, 6.13698867291239538994382008555, 7.46540798749994557457233480184, 8.067360394815936032060835518047, 9.022860087613765962016069408275