L(s) = 1 | + (0.723 − 0.690i)3-s + (0.415 + 0.909i)4-s + (−0.786 − 0.618i)7-s + (0.0475 − 0.998i)9-s + (0.928 + 0.371i)12-s + (1.56 + 0.625i)13-s + (−0.654 + 0.755i)16-s + (0.396 − 0.254i)19-s + (−0.995 + 0.0950i)21-s + (0.928 + 0.371i)25-s + (−0.654 − 0.755i)27-s + (0.235 − 0.971i)28-s + (0.186 − 1.29i)31-s + (0.928 − 0.371i)36-s − 1.57·37-s + ⋯ |
L(s) = 1 | + (0.723 − 0.690i)3-s + (0.415 + 0.909i)4-s + (−0.786 − 0.618i)7-s + (0.0475 − 0.998i)9-s + (0.928 + 0.371i)12-s + (1.56 + 0.625i)13-s + (−0.654 + 0.755i)16-s + (0.396 − 0.254i)19-s + (−0.995 + 0.0950i)21-s + (0.928 + 0.371i)25-s + (−0.654 − 0.755i)27-s + (0.235 − 0.971i)28-s + (0.186 − 1.29i)31-s + (0.928 − 0.371i)36-s − 1.57·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.470340036\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.470340036\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.723 + 0.690i)T \) |
| 7 | \( 1 + (0.786 + 0.618i)T \) |
| 67 | \( 1 + (0.786 - 0.618i)T \) |
good | 2 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 5 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 11 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 13 | \( 1 + (-1.56 - 0.625i)T + (0.723 + 0.690i)T^{2} \) |
| 17 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 19 | \( 1 + (-0.396 + 0.254i)T + (0.415 - 0.909i)T^{2} \) |
| 23 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \) |
| 37 | \( 1 + 1.57T + T^{2} \) |
| 41 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 43 | \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \) |
| 47 | \( 1 + (-0.0475 + 0.998i)T^{2} \) |
| 53 | \( 1 + (-0.981 - 0.189i)T^{2} \) |
| 59 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 61 | \( 1 + (0.308 + 0.356i)T + (-0.142 + 0.989i)T^{2} \) |
| 71 | \( 1 + (0.327 + 0.945i)T^{2} \) |
| 73 | \( 1 + (0.642 - 1.85i)T + (-0.786 - 0.618i)T^{2} \) |
| 79 | \( 1 + (1.45 + 0.584i)T + (0.723 + 0.690i)T^{2} \) |
| 83 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 89 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 97 | \( 1 + (0.723 - 1.25i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.434178332082661928631994514852, −8.783056351249595292996107804507, −8.120302925825081690631685003702, −7.17821838808293972128318150671, −6.76840833498785337017485904096, −5.95174419969584695148872774657, −4.18562216923995715454691823730, −3.52454819097518152112372555188, −2.76233827813460515813571684459, −1.42986227353894751810014168920,
1.56181742358937458308982330241, 2.88175185648265899783932716200, 3.50968258410974300799801610622, 4.84849367212256253117415082064, 5.65030485715970432359525427710, 6.38131807894816025540260054823, 7.32872177254567904962282173280, 8.657920331221584426653593327366, 8.824069701061931715559752884752, 9.904076965156001015053987352768