L(s) = 1 | + (0.707 + 0.707i)2-s + (0.352 + 1.69i)3-s + 1.00i·4-s + (−0.499 − 0.499i)5-s + (−0.949 + 1.44i)6-s + (−1.39 − 1.39i)7-s + (−0.707 + 0.707i)8-s + (−2.75 + 1.19i)9-s − 0.705i·10-s + (−3.39 + 3.39i)11-s + (−1.69 + 0.352i)12-s − 1.97i·14-s + (0.670 − 1.02i)15-s − 1.00·16-s − 4.38·17-s + (−2.79 − 1.09i)18-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.203 + 0.979i)3-s + 0.500i·4-s + (−0.223 − 0.223i)5-s + (−0.387 + 0.591i)6-s + (−0.528 − 0.528i)7-s + (−0.250 + 0.250i)8-s + (−0.916 + 0.398i)9-s − 0.223i·10-s + (−1.02 + 1.02i)11-s + (−0.489 + 0.101i)12-s − 0.528i·14-s + (0.173 − 0.263i)15-s − 0.250·16-s − 1.06·17-s + (−0.657 − 0.259i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 + 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.767 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7287589730\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7287589730\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.352 - 1.69i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + (0.499 + 0.499i)T + 5iT^{2} \) |
| 7 | \( 1 + (1.39 + 1.39i)T + 7iT^{2} \) |
| 11 | \( 1 + (3.39 - 3.39i)T - 11iT^{2} \) |
| 17 | \( 1 + 4.38T + 17T^{2} \) |
| 19 | \( 1 + (-1.70 + 1.70i)T - 19iT^{2} \) |
| 23 | \( 1 - 0.998T + 23T^{2} \) |
| 29 | \( 1 - 0.998iT - 29T^{2} \) |
| 31 | \( 1 + (6.50 - 6.50i)T - 31iT^{2} \) |
| 37 | \( 1 + (4.10 + 4.10i)T + 37iT^{2} \) |
| 41 | \( 1 + (-5.24 - 5.24i)T + 41iT^{2} \) |
| 43 | \( 1 - 8.88iT - 43T^{2} \) |
| 47 | \( 1 + (0.352 - 0.352i)T - 47iT^{2} \) |
| 53 | \( 1 - 14.2iT - 53T^{2} \) |
| 59 | \( 1 + (-0.998 + 0.998i)T - 59iT^{2} \) |
| 61 | \( 1 + 9.59T + 61T^{2} \) |
| 67 | \( 1 + (-5.79 + 5.79i)T - 67iT^{2} \) |
| 71 | \( 1 + (7.13 + 7.13i)T + 71iT^{2} \) |
| 73 | \( 1 + (-1.70 - 1.70i)T + 73iT^{2} \) |
| 79 | \( 1 + 0.207T + 79T^{2} \) |
| 83 | \( 1 + (-9.17 - 9.17i)T + 83iT^{2} \) |
| 89 | \( 1 + (-2.54 + 2.54i)T - 89iT^{2} \) |
| 97 | \( 1 + (-3.58 + 3.58i)T - 97iT^{2} \) |
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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
−10.51837542590854547216902920792, −9.537190629097432489493449257147, −8.890713295445387152864342933976, −7.86654227937522009822903609523, −7.15256464021678503690968195060, −6.12376421997132353936260744753, −4.94331727839183519207521483502, −4.54952633254237913824971892424, −3.49223497135434474538196890861, −2.47734595394730749128640337337,
0.24982158451112012885221036390, 2.00255853036477770416750878133, 2.91236993699083040535085492736, 3.71530577645530991042900417051, 5.33720956244941599177400810594, 5.89426874461424354677583052407, 6.86281197377835629861032740916, 7.71289087910827450650475225890, 8.667027853537051549920799499623, 9.334899314662406697538059460328