L(s) = 1 | + (−0.959 + 0.281i)2-s + (−1.07 + 1.66i)3-s + (0.841 − 0.540i)4-s + (0.557 − 1.89i)6-s + (−0.654 + 0.755i)8-s + (−1.21 − 2.65i)9-s + (−0.857 − 0.989i)11-s + 1.97i·12-s + (0.415 − 0.909i)16-s + (−1.84 − 0.540i)17-s + (1.91 + 2.20i)18-s + (−1.37 + 0.627i)19-s + (1.10 + 0.708i)22-s + (−0.557 − 1.89i)24-s + (−0.142 − 0.989i)25-s + ⋯ |
L(s) = 1 | + (−0.959 + 0.281i)2-s + (−1.07 + 1.66i)3-s + (0.841 − 0.540i)4-s + (0.557 − 1.89i)6-s + (−0.654 + 0.755i)8-s + (−1.21 − 2.65i)9-s + (−0.857 − 0.989i)11-s + 1.97i·12-s + (0.415 − 0.909i)16-s + (−1.84 − 0.540i)17-s + (1.91 + 2.20i)18-s + (−1.37 + 0.627i)19-s + (1.10 + 0.708i)22-s + (−0.557 − 1.89i)24-s + (−0.142 − 0.989i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0928 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0928 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.05726893014\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05726893014\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 89 | \( 1 + (0.142 - 0.989i)T \) |
good | 3 | \( 1 + (1.07 - 1.66i)T + (-0.415 - 0.909i)T^{2} \) |
| 5 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 7 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 11 | \( 1 + (0.857 + 0.989i)T + (-0.142 + 0.989i)T^{2} \) |
| 13 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 17 | \( 1 + (1.84 + 0.540i)T + (0.841 + 0.540i)T^{2} \) |
| 19 | \( 1 + (1.37 - 0.627i)T + (0.654 - 0.755i)T^{2} \) |
| 23 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 29 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 31 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 43 | \( 1 + (0.425 - 0.368i)T + (0.142 - 0.989i)T^{2} \) |
| 47 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 53 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 59 | \( 1 + (0.304 + 0.474i)T + (-0.415 + 0.909i)T^{2} \) |
| 61 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 67 | \( 1 + (0.698 + 0.449i)T + (0.415 + 0.909i)T^{2} \) |
| 71 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 73 | \( 1 + (-0.118 + 0.258i)T + (-0.654 - 0.755i)T^{2} \) |
| 79 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 83 | \( 1 + (0.304 - 1.03i)T + (-0.841 - 0.540i)T^{2} \) |
| 97 | \( 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2} \) |
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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
−10.59500937301234215168451258406, −9.661139207139599018179503207851, −8.879994929428453489030577941061, −8.226798612440297171012698555028, −6.58176149947625357772347710532, −6.07757305683537156651168853152, −5.10380817578331401381502902163, −4.16941828939288576220831039761, −2.72847182945239707993008912618, −0.087720807979008825358061568418,
1.80061388911622004169962072277, 2.42977696132648258764099142105, 4.64166250587002288280486906020, 5.89609267485123009301570663358, 6.80944904921304737190291428071, 7.19016547691413003394056010945, 8.164211893910006635064352234615, 8.873951312493918146170227296452, 10.33026485882329984077591034794, 10.90669541640051442339804043802