L(s) = 1 | + (0.707 + 0.707i)2-s + (−1.63 + 0.577i)3-s + 1.00i·4-s + (−2.23 − 0.0135i)5-s + (−1.56 − 0.746i)6-s + (3.38 − 3.38i)7-s + (−0.707 + 0.707i)8-s + (2.33 − 1.88i)9-s + (−1.57 − 1.59i)10-s − 1.22i·11-s + (−0.577 − 1.63i)12-s + (3.51 + 3.51i)13-s + 4.78·14-s + (3.65 − 1.26i)15-s − 1.00·16-s + (−0.826 − 0.826i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (−0.942 + 0.333i)3-s + 0.500i·4-s + (−0.999 − 0.00606i)5-s + (−0.638 − 0.304i)6-s + (1.27 − 1.27i)7-s + (−0.250 + 0.250i)8-s + (0.777 − 0.628i)9-s + (−0.496 − 0.503i)10-s − 0.369i·11-s + (−0.166 − 0.471i)12-s + (0.975 + 0.975i)13-s + 1.27·14-s + (0.944 − 0.327i)15-s − 0.250·16-s + (−0.200 − 0.200i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.34656 + 0.391404i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.34656 + 0.391404i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (1.63 - 0.577i)T \) |
| 5 | \( 1 + (2.23 + 0.0135i)T \) |
| 19 | \( 1 - iT \) |
good | 7 | \( 1 + (-3.38 + 3.38i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.22iT - 11T^{2} \) |
| 13 | \( 1 + (-3.51 - 3.51i)T + 13iT^{2} \) |
| 17 | \( 1 + (0.826 + 0.826i)T + 17iT^{2} \) |
| 23 | \( 1 + (-1.04 + 1.04i)T - 23iT^{2} \) |
| 29 | \( 1 - 8.14T + 29T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 + (5.83 - 5.83i)T - 37iT^{2} \) |
| 41 | \( 1 + 3.53iT - 41T^{2} \) |
| 43 | \( 1 + (2.68 + 2.68i)T + 43iT^{2} \) |
| 47 | \( 1 + (-6.84 - 6.84i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.40 + 3.40i)T - 53iT^{2} \) |
| 59 | \( 1 - 9.91T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + (-6.00 + 6.00i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.35iT - 71T^{2} \) |
| 73 | \( 1 + (3.99 + 3.99i)T + 73iT^{2} \) |
| 79 | \( 1 + 6.75iT - 79T^{2} \) |
| 83 | \( 1 + (0.717 - 0.717i)T - 83iT^{2} \) |
| 89 | \( 1 - 0.384T + 89T^{2} \) |
| 97 | \( 1 + (12.5 - 12.5i)T - 97iT^{2} \) |
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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
−10.94401184598750347054792679427, −10.38278528211578586253257930492, −8.759323189607538299457262113031, −8.011794633090448220500269966374, −7.03299187060531984032957034974, −6.39674657002658770853714582693, −4.92935678683024736238929411351, −4.41981481313099477892954796035, −3.63230247103950802439977834198, −1.05169450331380853135380840907,
1.17730119146007592173369824526, 2.64853742559141296346331245182, 4.21213950802839091905698722353, 5.05966657190350221504565036350, 5.78532955480646902860082531509, 6.92521742538708542362451272481, 8.117830994564349738169866912698, 8.647563342367019568420058119176, 10.25438589979297509471706260394, 10.99297341382999966797147204213