L(s) = 1 | + (−0.618 + 1.27i)2-s + (−2.16 − 2.16i)3-s + (−1.23 − 1.57i)4-s + (4.09 − 1.41i)6-s − 3.30i·7-s + (2.76 − 0.594i)8-s + 6.40i·9-s + (2.01 − 2.01i)11-s + (−0.738 + 6.08i)12-s + (−0.794 − 0.794i)13-s + (4.20 + 2.04i)14-s + (−0.955 + 3.88i)16-s − 4.61·17-s + (−8.14 − 3.96i)18-s + (−3.48 − 3.48i)19-s + ⋯ |
L(s) = 1 | + (−0.437 + 0.899i)2-s + (−1.25 − 1.25i)3-s + (−0.616 − 0.787i)4-s + (1.67 − 0.577i)6-s − 1.24i·7-s + (0.977 − 0.210i)8-s + 2.13i·9-s + (0.606 − 0.606i)11-s + (−0.213 + 1.75i)12-s + (−0.220 − 0.220i)13-s + (1.12 + 0.546i)14-s + (−0.238 + 0.971i)16-s − 1.11·17-s + (−1.91 − 0.934i)18-s + (−0.800 − 0.800i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.150i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0186862 - 0.246309i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0186862 - 0.246309i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.618 - 1.27i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (2.16 + 2.16i)T + 3iT^{2} \) |
| 7 | \( 1 + 3.30iT - 7T^{2} \) |
| 11 | \( 1 + (-2.01 + 2.01i)T - 11iT^{2} \) |
| 13 | \( 1 + (0.794 + 0.794i)T + 13iT^{2} \) |
| 17 | \( 1 + 4.61T + 17T^{2} \) |
| 19 | \( 1 + (3.48 + 3.48i)T + 19iT^{2} \) |
| 23 | \( 1 - 7.99iT - 23T^{2} \) |
| 29 | \( 1 + (1.95 + 1.95i)T + 29iT^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 + (0.448 - 0.448i)T - 37iT^{2} \) |
| 41 | \( 1 - 4.02iT - 41T^{2} \) |
| 43 | \( 1 + (4.97 - 4.97i)T - 43iT^{2} \) |
| 47 | \( 1 - 5.49T + 47T^{2} \) |
| 53 | \( 1 + (-3.35 + 3.35i)T - 53iT^{2} \) |
| 59 | \( 1 + (-2.07 + 2.07i)T - 59iT^{2} \) |
| 61 | \( 1 + (0.557 + 0.557i)T + 61iT^{2} \) |
| 67 | \( 1 + (-0.636 - 0.636i)T + 67iT^{2} \) |
| 71 | \( 1 + 6.85iT - 71T^{2} \) |
| 73 | \( 1 - 10.5iT - 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 + (9.48 + 9.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 7.62iT - 89T^{2} \) |
| 97 | \( 1 + 0.709T + 97T^{2} \) |
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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
−11.10378102175907031659188226557, −10.00437604569859386198862938164, −8.747998624998573747397204577961, −7.60505166630629139500826702000, −7.03216608026882429990380623980, −6.34914891030751978247856030774, −5.41129258765050148914429605831, −4.24230220443140480653966579587, −1.50807533695997965112678241459, −0.22506274228281940786904113730,
2.22046163796691435443339522494, 3.92244566904294478980721855294, 4.66119137112476918290945834040, 5.71777974159860096060834224370, 6.84061454147350983796178018867, 8.727748687622477509844888597671, 9.094428491616133672667161659239, 10.14699863853397762443232381498, 10.72902639220147760271523181264, 11.60386894824410470410279061399