L(s) = 1 | + (0.707 + 0.707i)2-s + (0.923 + 0.382i)3-s + 1.00i·4-s + (−0.923 − 0.382i)5-s + (0.382 + 0.923i)6-s + (−0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.382 − 0.923i)10-s + (−0.382 + 0.923i)12-s + (0.541 + 0.541i)13-s − 1.00·14-s + (−0.707 − 0.707i)15-s − 1.00·16-s + i·18-s − 1.84i·19-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + (0.923 + 0.382i)3-s + 1.00i·4-s + (−0.923 − 0.382i)5-s + (0.382 + 0.923i)6-s + (−0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.382 − 0.923i)10-s + (−0.382 + 0.923i)12-s + (0.541 + 0.541i)13-s − 1.00·14-s + (−0.707 − 0.707i)15-s − 1.00·16-s + i·18-s − 1.84i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.525658252\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.525658252\) |
\(L(1)\) |
|
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 | \( 1 + (-0.923 - 0.382i)T \) |
| 5 | \( 1 + (0.923 + 0.382i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + 1.84iT - T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + 1.84T + T^{2} \) |
| 61 | \( 1 - 0.765T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−10.78585464720617963781201474139, −9.155466524746324999096063365647, −9.000014566274115645308147978887, −8.151749942732306014185869971723, −7.17439059037770125231148667983, −6.48639996496288080123107432743, −5.06682508792269630662216475144, −4.41911498370607135580904484895, −3.39360023381997439146620599089, −2.61627501555961656032148228511,
1.32865907606311989041910774252, 2.98122520361541968056693347761, 3.54891342543607918975105032350, 4.24286165532814368269559004282, 5.78708767623472826715170287234, 6.78267052250107349946304468899, 7.54516678965835123501355622276, 8.440838059413033704745552512358, 9.542679093651995019120629520906, 10.24814765369641652022554755161