L(s) = 1 | + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s + (0.587 − 0.190i)13-s + (0.309 + 0.951i)16-s + (0.951 + 1.30i)17-s − 0.999i·18-s + 0.618·26-s + (−1.30 − 0.951i)29-s + i·32-s + (0.499 + 1.53i)34-s + (0.309 − 0.951i)36-s + (1.53 − 0.5i)37-s + (0.190 + 0.587i)41-s + ⋯ |
L(s) = 1 | + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s + (0.587 − 0.190i)13-s + (0.309 + 0.951i)16-s + (0.951 + 1.30i)17-s − 0.999i·18-s + 0.618·26-s + (−1.30 − 0.951i)29-s + i·32-s + (0.499 + 1.53i)34-s + (0.309 − 0.951i)36-s + (1.53 − 0.5i)37-s + (0.190 + 0.587i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.247843640\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.247843640\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.951 - 0.309i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
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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
−9.106516237197797618422824147848, −8.100084456902377755872647803646, −7.71354421216587171413924538263, −6.54623312504236163957111081702, −6.03599250601619770518332428976, −5.46511811082813290352176007448, −4.21397714255248979253885898440, −3.68114378799668163629428273763, −2.81201456969045353424007726141, −1.49698339769286531086402846576,
1.37358505941803403156758882708, 2.51156753555057922610181773711, 3.29322901309741937730363633878, 4.26856163391458613995349159481, 5.18396476268936980214004123645, 5.62171261080230465408455876826, 6.62442109036484551620187371774, 7.43135419430000842565071412971, 8.059490514428628971290095254255, 9.212005797763672034787448451084