L(s) = 1 | + i·2-s − 4-s − i·8-s + i·13-s + 16-s + 6i·17-s − 4i·23-s − 26-s − 10·29-s + i·32-s − 6·34-s − 6i·37-s − 2·41-s + 4i·43-s + 4·46-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.353i·8-s + 0.277i·13-s + 0.250·16-s + 1.45i·17-s − 0.834i·23-s − 0.196·26-s − 1.85·29-s + 0.176i·32-s − 1.02·34-s − 0.986i·37-s − 0.312·41-s + 0.609i·43-s + 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−7.79731087266028861009681692570, −7.19230858787270129043354667308, −6.42028226033848979622776635160, −5.82204123280065526217051617247, −5.15183336754910328303425910429, −4.12622634366027146544725128692, −3.73343628683964628883440214156, −2.45533997525300479164022074632, −1.45869179592622687129057311648, 0,
1.20078728264625669791527588596, 2.21239963657235457594495411328, 3.06533540760085664296882875957, 3.75810668214042934493652976928, 4.67300440398795880844682225794, 5.36598938657312857091367500455, 6.00822673083805806939131080129, 7.20280929015666508066317455319, 7.49004487858743812481928371522