L(s) = 1 | + (−0.707 + 0.707i)3-s + (−0.707 + 0.707i)5-s + (0.707 + 0.707i)7-s − 1.00i·9-s − 1.00i·15-s + (1.41 − 1.41i)17-s + 1.41·19-s − 1.00·21-s + (1 − i)23-s − 1.00i·25-s + (0.707 + 0.707i)27-s + 1.41i·31-s − 1.00·35-s + (1 − i)37-s − 1.41·41-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)3-s + (−0.707 + 0.707i)5-s + (0.707 + 0.707i)7-s − 1.00i·9-s − 1.00i·15-s + (1.41 − 1.41i)17-s + 1.41·19-s − 1.00·21-s + (1 − i)23-s − 1.00i·25-s + (0.707 + 0.707i)27-s + 1.41i·31-s − 1.00·35-s + (1 − i)37-s − 1.41·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.036190101\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.036190101\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 19 | \( 1 - 1.41T + T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - 1.41iT - T^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - 2iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 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
−8.986769336380009530665503794298, −8.179160254785251011303595656304, −7.30029736544056758020901346589, −6.80362973531138789033787796906, −5.63334491364519091240691944569, −5.18028170404715425964326431655, −4.44511167527059901729445315724, −3.29859349211270150538738217531, −2.82051512212679484165847567451, −1.01967869029384785975645187616,
1.01305516357556828690413184279, 1.56896874111692806635903628188, 3.25344723551540963051324191330, 4.09291264129803258365908783732, 5.07397893881095112942983844717, 5.46862014067402229555304726118, 6.46960342381366116494048825295, 7.48864980641307776032480220716, 7.77997601204845604305511038232, 8.299542090261446822770517052834