L(s) = 1 | + 1.61·2-s + 3-s + 0.618·4-s + 1.61·6-s + 4.61·7-s − 2.23·8-s + 9-s − 2.23·11-s + 0.618·12-s + 1.76·13-s + 7.47·14-s − 4.85·16-s + 4.85·17-s + 1.61·18-s − 8.09·19-s + 4.61·21-s − 3.61·22-s + 2.38·23-s − 2.23·24-s + 2.85·26-s + 27-s + 2.85·28-s + 8.61·29-s + 9.56·31-s − 3.38·32-s − 2.23·33-s + 7.85·34-s + ⋯ |
L(s) = 1 | + 1.14·2-s + 0.577·3-s + 0.309·4-s + 0.660·6-s + 1.74·7-s − 0.790·8-s + 0.333·9-s − 0.674·11-s + 0.178·12-s + 0.489·13-s + 1.99·14-s − 1.21·16-s + 1.17·17-s + 0.381·18-s − 1.85·19-s + 1.00·21-s − 0.771·22-s + 0.496·23-s − 0.456·24-s + 0.559·26-s + 0.192·27-s + 0.539·28-s + 1.60·29-s + 1.71·31-s − 0.597·32-s − 0.389·33-s + 1.34·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.959188947\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.959188947\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 59 | \( 1 + T \) |
good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 7 | \( 1 - 4.61T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 + 8.09T + 19T^{2} \) |
| 23 | \( 1 - 2.38T + 23T^{2} \) |
| 29 | \( 1 - 8.61T + 29T^{2} \) |
| 31 | \( 1 - 9.56T + 31T^{2} \) |
| 37 | \( 1 - 6.85T + 37T^{2} \) |
| 41 | \( 1 + 3.09T + 41T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 - 4.14T + 47T^{2} \) |
| 53 | \( 1 + 1.76T + 53T^{2} \) |
| 61 | \( 1 + 9.85T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 - 9.94T + 71T^{2} \) |
| 73 | \( 1 - 5.85T + 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 + 0.618T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 3T + 97T^{2} \) |
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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
−8.252419121729963466604057602243, −7.895568097799451998359497492915, −6.70631292813660061906677071970, −6.01550206942327817836940518226, −5.06420405368235833901953968529, −4.65393605379834373742175500608, −4.01903409892236909916379294999, −2.96426507038382580072113218299, −2.28438336394772667286516783471, −1.10115142098212647082118293732,
1.10115142098212647082118293732, 2.28438336394772667286516783471, 2.96426507038382580072113218299, 4.01903409892236909916379294999, 4.65393605379834373742175500608, 5.06420405368235833901953968529, 6.01550206942327817836940518226, 6.70631292813660061906677071970, 7.895568097799451998359497492915, 8.252419121729963466604057602243