L(s) = 1 | + (−0.928 + 0.371i)2-s + (0.947 + 1.09i)3-s + (0.723 − 0.690i)4-s + (0.841 + 0.540i)5-s + (−1.28 − 0.663i)6-s + (−0.514 − 0.404i)7-s + (−0.415 + 0.909i)8-s + (−0.155 + 1.08i)9-s + (−0.981 − 0.189i)10-s + (1.44 + 0.137i)12-s + (0.627 + 0.184i)14-s + (0.205 + 1.43i)15-s + (0.0475 − 0.998i)16-s + (−0.258 − 1.06i)18-s + (0.981 − 0.189i)20-s + (−0.0450 − 0.945i)21-s + ⋯ |
L(s) = 1 | + (−0.928 + 0.371i)2-s + (0.947 + 1.09i)3-s + (0.723 − 0.690i)4-s + (0.841 + 0.540i)5-s + (−1.28 − 0.663i)6-s + (−0.514 − 0.404i)7-s + (−0.415 + 0.909i)8-s + (−0.155 + 1.08i)9-s + (−0.981 − 0.189i)10-s + (1.44 + 0.137i)12-s + (0.627 + 0.184i)14-s + (0.205 + 1.43i)15-s + (0.0475 − 0.998i)16-s + (−0.258 − 1.06i)18-s + (0.981 − 0.189i)20-s + (−0.0450 − 0.945i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.056101330\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.056101330\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.928 - 0.371i)T \) |
| 5 | \( 1 + (-0.841 - 0.540i)T \) |
| 67 | \( 1 + (0.235 + 0.971i)T \) |
good | 3 | \( 1 + (-0.947 - 1.09i)T + (-0.142 + 0.989i)T^{2} \) |
| 7 | \( 1 + (0.514 + 0.404i)T + (0.235 + 0.971i)T^{2} \) |
| 11 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 13 | \( 1 + (0.327 + 0.945i)T^{2} \) |
| 17 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 19 | \( 1 + (-0.235 + 0.971i)T^{2} \) |
| 23 | \( 1 + (0.327 - 0.945i)T + (-0.786 - 0.618i)T^{2} \) |
| 29 | \( 1 + (0.981 - 1.70i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.273 + 1.12i)T + (-0.888 - 0.458i)T^{2} \) |
| 43 | \( 1 + (-0.452 + 0.132i)T + (0.841 - 0.540i)T^{2} \) |
| 47 | \( 1 + (-1.95 + 0.376i)T + (0.928 - 0.371i)T^{2} \) |
| 53 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 59 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 61 | \( 1 + (1.49 + 0.770i)T + (0.580 + 0.814i)T^{2} \) |
| 71 | \( 1 + (-0.0475 + 0.998i)T^{2} \) |
| 73 | \( 1 + (-0.580 - 0.814i)T^{2} \) |
| 79 | \( 1 + (-0.981 - 0.189i)T^{2} \) |
| 83 | \( 1 + (-0.0748 + 1.57i)T + (-0.995 - 0.0950i)T^{2} \) |
| 89 | \( 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−9.823695318583474233487975340441, −9.221503414530376650545615460087, −8.838527773886037484090754619184, −7.62854935930050477916549023373, −7.01117518266900097902647913982, −5.98056529175461133937868609308, −5.16216366605543521266200387338, −3.74968589787967634191178070135, −2.95180229866564744406562298617, −1.83843926789901982320900795172,
1.16276497683607374474389941344, 2.30482106585476922562708740002, 2.73642430919656068769581916318, 4.17423726255157274003290411320, 5.87594878176282160926876090083, 6.45023234328095257293406434714, 7.45950065590377298599665753795, 8.087951001731615730398391744332, 8.868007028274146832211983116418, 9.357630255618278097790767441107