L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 3.69·7-s + 8-s + 9-s − 10-s − 3.04·11-s + 12-s + 3.69·14-s − 15-s + 16-s − 6.85·17-s + 18-s + 0.911·19-s − 20-s + 3.69·21-s − 3.04·22-s − 0.356·23-s + 24-s + 25-s + 27-s + 3.69·28-s + 10.5·29-s − 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 1.39·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.919·11-s + 0.288·12-s + 0.986·14-s − 0.258·15-s + 0.250·16-s − 1.66·17-s + 0.235·18-s + 0.209·19-s − 0.223·20-s + 0.805·21-s − 0.650·22-s − 0.0744·23-s + 0.204·24-s + 0.200·25-s + 0.192·27-s + 0.697·28-s + 1.95·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.143389025\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.143389025\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 3.69T + 7T^{2} \) |
| 11 | \( 1 + 3.04T + 11T^{2} \) |
| 17 | \( 1 + 6.85T + 17T^{2} \) |
| 19 | \( 1 - 0.911T + 19T^{2} \) |
| 23 | \( 1 + 0.356T + 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 - 2.06T + 31T^{2} \) |
| 37 | \( 1 + 0.899T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 9.43T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 - 8.54T + 59T^{2} \) |
| 61 | \( 1 - 1.55T + 61T^{2} \) |
| 67 | \( 1 - 1.14T + 67T^{2} \) |
| 71 | \( 1 + 4.13T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 9.62T + 79T^{2} \) |
| 83 | \( 1 + 9.86T + 83T^{2} \) |
| 89 | \( 1 + 6.41T + 89T^{2} \) |
| 97 | \( 1 - 6.97T + 97T^{2} \) |
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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
−8.210442758138017058202961202277, −7.54318631181925867240888082226, −6.91828353460135478628495402233, −5.96828891131067998743911088657, −5.04304401098518053194329168294, −4.49901256574526282666274096927, −3.97251512219645676968407269603, −2.63766328641122596027233868394, −2.33423224006432297351616982705, −1.00203575540890503852424381996,
1.00203575540890503852424381996, 2.33423224006432297351616982705, 2.63766328641122596027233868394, 3.97251512219645676968407269603, 4.49901256574526282666274096927, 5.04304401098518053194329168294, 5.96828891131067998743911088657, 6.91828353460135478628495402233, 7.54318631181925867240888082226, 8.210442758138017058202961202277