L(s) = 1 | + i·2-s + 3-s − 4-s + i·5-s + i·6-s − 2.24i·7-s − i·8-s + 9-s − 10-s − 1.69i·11-s − 12-s + 2.24·14-s + i·15-s + 16-s − 0.445·17-s + i·18-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s + 0.408i·6-s − 0.849i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s − 0.510i·11-s − 0.288·12-s + 0.600·14-s + 0.258i·15-s + 0.250·16-s − 0.107·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.613892960\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.613892960\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 2.24iT - 7T^{2} \) |
| 11 | \( 1 + 1.69iT - 11T^{2} \) |
| 17 | \( 1 + 0.445T + 17T^{2} \) |
| 19 | \( 1 + 1.74iT - 19T^{2} \) |
| 23 | \( 1 + 4.04T + 23T^{2} \) |
| 29 | \( 1 - 0.643T + 29T^{2} \) |
| 31 | \( 1 - 4.80iT - 31T^{2} \) |
| 37 | \( 1 + 0.862iT - 37T^{2} \) |
| 41 | \( 1 + 5.74iT - 41T^{2} \) |
| 43 | \( 1 - 1.58T + 43T^{2} \) |
| 47 | \( 1 - 5.49iT - 47T^{2} \) |
| 53 | \( 1 - 0.137T + 53T^{2} \) |
| 59 | \( 1 + 12.6iT - 59T^{2} \) |
| 61 | \( 1 + 0.472T + 61T^{2} \) |
| 67 | \( 1 + 4.54iT - 67T^{2} \) |
| 71 | \( 1 + 12.6iT - 71T^{2} \) |
| 73 | \( 1 + 11.1iT - 73T^{2} \) |
| 79 | \( 1 + 6.57T + 79T^{2} \) |
| 83 | \( 1 + 1.91iT - 83T^{2} \) |
| 89 | \( 1 + 11.9iT - 89T^{2} \) |
| 97 | \( 1 + 7.55iT - 97T^{2} \) |
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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
−7.898098976591517039096524584513, −7.54897079637246913304188355040, −6.73039894992897190985273965057, −6.21630271410665662153240882717, −5.24157552910289266012948692927, −4.38480677636047429615699545991, −3.67932824773429239189129645546, −2.94527732800480714358800559275, −1.74763689616276904540935217587, −0.40890069576058870502851439435,
1.20304211082827295795677161140, 2.18228403429936001327572603428, 2.72964872658782798221365078219, 3.86299703256792629377356151257, 4.38349986688549289342259580702, 5.35802809610884312101036821952, 5.95891377132161671882783804823, 7.00849887683437725629498778132, 7.87737785981162074969986942306, 8.476372541875727384344017482701