| L(s) = 1 | + (1.32 + 0.5i)2-s + (1.50 + 1.32i)4-s − 2.64·7-s + (1.32 + 2.50i)8-s + 3·9-s + 5.29i·11-s + (−3.50 − 1.32i)14-s + (0.500 + 3.96i)16-s + (3.96 + 1.5i)18-s + (−2.64 + 7.00i)22-s + 5.29·23-s + (−3.96 − 3.50i)28-s + 2·29-s + (−1.32 + 5.50i)32-s + (4.50 + 3.96i)36-s − 6i·37-s + ⋯ |
| L(s) = 1 | + (0.935 + 0.353i)2-s + (0.750 + 0.661i)4-s − 0.999·7-s + (0.467 + 0.883i)8-s + 9-s + 1.59i·11-s + (−0.935 − 0.353i)14-s + (0.125 + 0.992i)16-s + (0.935 + 0.353i)18-s + (−0.564 + 1.49i)22-s + 1.10·23-s + (−0.749 − 0.661i)28-s + 0.371·29-s + (−0.233 + 0.972i)32-s + (0.750 + 0.661i)36-s − 0.986i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.03171 + 1.56337i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.03171 + 1.56337i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.32 - 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 11 | \( 1 - 5.29iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 + 5.29iT - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 15.8iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−10.55952511486897903080040795509, −9.893281636433820696428973352961, −8.950679828292067419215354381841, −7.53168855007886433463883966674, −7.05076627576854977036085459566, −6.29454097402071341844582926550, −5.03433084705115989692616454275, −4.28663699506052267469562859014, −3.22783224177578764537733490892, −1.93832647899522068186978320012,
1.09044148761920398718652922031, 2.84313355896016259220243557796, 3.57291832868904654165241489900, 4.66765099047661922255055650159, 5.78959858919623842331192490020, 6.52524392872284798905455706899, 7.32144003156263686678665776569, 8.658791757311803904222507445341, 9.664224962318890664126878757503, 10.42129179336290594647972038787
