L(s) = 1 | + 2.56·3-s + (2 − i)5-s + (2 + 1.73i)7-s + 3.56·9-s + 0.972·11-s + 0.561i·13-s + (5.12 − 2.56i)15-s + 4.43·17-s − 1.12i·19-s + (5.12 + 4.43i)21-s − 1.12·23-s + (3 − 4i)25-s + 1.43·27-s + 4.43i·29-s − 8.87·31-s + ⋯ |
L(s) = 1 | + 1.47·3-s + (0.894 − 0.447i)5-s + (0.755 + 0.654i)7-s + 1.18·9-s + 0.293·11-s + 0.155i·13-s + (1.32 − 0.661i)15-s + 1.07·17-s − 0.257i·19-s + (1.11 + 0.968i)21-s − 0.234·23-s + (0.600 − 0.800i)25-s + 0.276·27-s + 0.823i·29-s − 1.59·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.994607849\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.994607849\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2 + i)T \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 11 | \( 1 - 0.972T + 11T^{2} \) |
| 13 | \( 1 - 0.561iT - 13T^{2} \) |
| 17 | \( 1 - 4.43T + 17T^{2} \) |
| 19 | \( 1 + 1.12iT - 19T^{2} \) |
| 23 | \( 1 + 1.12T + 23T^{2} \) |
| 29 | \( 1 - 4.43iT - 29T^{2} \) |
| 31 | \( 1 + 8.87T + 31T^{2} \) |
| 37 | \( 1 + 8.87T + 37T^{2} \) |
| 41 | \( 1 + 8.87iT - 41T^{2} \) |
| 43 | \( 1 - 1.94iT - 43T^{2} \) |
| 47 | \( 1 + 0.972iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 1.12T + 61T^{2} \) |
| 67 | \( 1 + 6.92iT - 67T^{2} \) |
| 71 | \( 1 - 14.2iT - 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 + 7.68iT - 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 8.87iT - 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
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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
−8.931784513850639222381421570274, −8.528130078345328748798194455740, −7.69071746556544338300132137741, −6.91343999247482935698539310779, −5.66430912149189491276316206711, −5.17374381626401831232743129904, −3.99899017013241652148913240542, −3.09772978969580001815871846796, −2.10560302853570763470627583641, −1.50410569257963854866895219684,
1.43343167222979505256879583295, 2.15506666234277472069332183487, 3.24418735291243727410256449906, 3.83599787333842979135375407084, 5.00883584450815888392809050151, 5.90280137347882942799465921397, 6.96344614866545930705661919857, 7.64231315280303524413960529837, 8.214848203419911573724880956876, 9.050376974303153973612729053301