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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.090443514\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.090443514\) |
\(L(1)\) |
|
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
−10.29198477150290416881528017755, −9.500518223528683895144411140050, −9.166842189676489370756360337842, −6.99661167159925464662231527160, −6.59651881563976812644538309588, −5.45139341551956633425090814006, −4.87850257892543722755341986590, −3.92403855270776471374600829562, −2.63594761260926485056462840590, −1.04836977250753396050973636224,
2.17313963422584616178933671452, 3.30468343959032288589146709290, 4.80605639466274270284233827658, 5.64561412159993957916347406893, 6.14959697885712239886420698928, 6.87064167453672881003413752102, 7.78361210686516619440327198340, 8.872841151987072751997676118566, 9.940472795476979361323315927512, 10.61528762471262184299128679657