L(s) = 1 | + (0.707 − 0.707i)3-s + i·5-s − 1.41·7-s + (−0.707 + 0.707i)11-s + i·13-s + (0.707 + 0.707i)15-s + (−1.41 + 1.41i)19-s + (−1.00 + 1.00i)21-s + (0.707 + 0.707i)27-s + 29-s + (0.707 − 0.707i)31-s + 1.00i·33-s − 1.41i·35-s + (0.707 + 0.707i)39-s + (−1 − i)41-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)3-s + i·5-s − 1.41·7-s + (−0.707 + 0.707i)11-s + i·13-s + (0.707 + 0.707i)15-s + (−1.41 + 1.41i)19-s + (−1.00 + 1.00i)21-s + (0.707 + 0.707i)27-s + 29-s + (0.707 − 0.707i)31-s + 1.00i·33-s − 1.41i·35-s + (0.707 + 0.707i)39-s + (−1 − i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9835920297\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9835920297\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 5 | \( 1 - iT - T^{2} \) |
| 7 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 13 | \( 1 - iT - T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (1 + i)T + iT^{2} \) |
| 43 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 47 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (1 - i)T - iT^{2} \) |
| 97 | \( 1 + (-1 - i)T + iT^{2} \) |
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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
−9.649159514398888120036266002659, −8.731696507392583136306916751918, −7.948709733229772297000421333855, −7.13608814725494379874283096038, −6.63077961046585948695637997221, −5.97771079787091627055071316688, −4.51869334095597338341397762853, −3.51324981420127589962644323853, −2.62392913839531244932858954037, −1.98946605598336116774647610982,
0.63402345575089894913957516964, 2.79080722367775947100230105479, 3.13490298660980465207957298132, 4.31528691210460824122775361291, 5.00899499753113726106202733436, 6.09203181661577622423280072603, 6.75369135790022347775392941003, 8.052983810240339784210172782302, 8.745541064418364848432576363055, 9.013727222080947155037382672189