L(s) = 1 | + 1.80·2-s − 1.89·3-s + 1.25·4-s − 5-s − 3.42·6-s − 7-s − 1.34·8-s + 0.600·9-s − 1.80·10-s − 1.17·11-s − 2.38·12-s − 4.31·13-s − 1.80·14-s + 1.89·15-s − 4.93·16-s − 6.27·17-s + 1.08·18-s − 8.34·19-s − 1.25·20-s + 1.89·21-s − 2.12·22-s − 2.43·23-s + 2.54·24-s + 25-s − 7.78·26-s + 4.55·27-s − 1.25·28-s + ⋯ |
L(s) = 1 | + 1.27·2-s − 1.09·3-s + 0.628·4-s − 0.447·5-s − 1.39·6-s − 0.377·7-s − 0.474·8-s + 0.200·9-s − 0.570·10-s − 0.355·11-s − 0.688·12-s − 1.19·13-s − 0.482·14-s + 0.489·15-s − 1.23·16-s − 1.52·17-s + 0.255·18-s − 1.91·19-s − 0.281·20-s + 0.414·21-s − 0.453·22-s − 0.508·23-s + 0.519·24-s + 0.200·25-s − 1.52·26-s + 0.876·27-s − 0.237·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.04990228483\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04990228483\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 - 1.80T + 2T^{2} \) |
| 3 | \( 1 + 1.89T + 3T^{2} \) |
| 11 | \( 1 + 1.17T + 11T^{2} \) |
| 13 | \( 1 + 4.31T + 13T^{2} \) |
| 17 | \( 1 + 6.27T + 17T^{2} \) |
| 19 | \( 1 + 8.34T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 + 8.45T + 29T^{2} \) |
| 31 | \( 1 - 6.67T + 31T^{2} \) |
| 37 | \( 1 + 8.35T + 37T^{2} \) |
| 41 | \( 1 - 3.15T + 41T^{2} \) |
| 43 | \( 1 + 4.66T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 5.67T + 53T^{2} \) |
| 59 | \( 1 - 7.57T + 59T^{2} \) |
| 61 | \( 1 + 6.87T + 61T^{2} \) |
| 67 | \( 1 + 9.26T + 67T^{2} \) |
| 71 | \( 1 + 4.02T + 71T^{2} \) |
| 73 | \( 1 + 8.17T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 + 3.42T + 89T^{2} \) |
| 97 | \( 1 + 7.01T + 97T^{2} \) |
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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
−7.46603208098944214681114082008, −6.80732060683828922876878851899, −6.26304531707153874878275359850, −5.68596280972189239714773290591, −4.94991802676441081043042507193, −4.41011905247896588853683711571, −3.89733500301104880539455027625, −2.74429670401553432639129139244, −2.15610354367591480785999334490, −0.087433182641929325070671098346,
0.087433182641929325070671098346, 2.15610354367591480785999334490, 2.74429670401553432639129139244, 3.89733500301104880539455027625, 4.41011905247896588853683711571, 4.94991802676441081043042507193, 5.68596280972189239714773290591, 6.26304531707153874878275359850, 6.80732060683828922876878851899, 7.46603208098944214681114082008