L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 1.04·7-s + 8-s + 9-s − 10-s + 1.35·11-s + 12-s − 1.04·14-s − 15-s + 16-s − 1.08·17-s + 18-s + 2.93·19-s − 20-s − 1.04·21-s + 1.35·22-s − 0.692·23-s + 24-s + 25-s + 27-s − 1.04·28-s − 2.37·29-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 − 0.396·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.409·11-s + 0.288·12-s − 0.280·14-s − 0.258·15-s + 0.250·16-s − 0.263·17-s + 0.235·18-s + 0.674·19-s − 0.223·20-s − 0.228·21-s + 0.289·22-s − 0.144·23-s + 0.204·24-s + 0.200·25-s + 0.192·27-s − 0.198·28-s − 0.441·29-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
\(L(1)\) |
\(\approx\) |
\(3.654814536\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.654814536\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.04T + 7T^{2} \) |
| 11 | \( 1 - 1.35T + 11T^{2} \) |
| 17 | \( 1 + 1.08T + 17T^{2} \) |
| 19 | \( 1 - 2.93T + 19T^{2} \) |
| 23 | \( 1 + 0.692T + 23T^{2} \) |
| 29 | \( 1 + 2.37T + 29T^{2} \) |
| 31 | \( 1 - 9.85T + 31T^{2} \) |
| 37 | \( 1 - 9.26T + 37T^{2} \) |
| 41 | \( 1 - 2.84T + 41T^{2} \) |
| 43 | \( 1 + 4.45T + 43T^{2} \) |
| 47 | \( 1 + 3.31T + 47T^{2} \) |
| 53 | \( 1 - 0.664T + 53T^{2} \) |
| 59 | \( 1 + 1.96T + 59T^{2} \) |
| 61 | \( 1 - 3.24T + 61T^{2} \) |
| 67 | \( 1 - 6.91T + 67T^{2} \) |
| 71 | \( 1 - 2.29T + 71T^{2} \) |
| 73 | \( 1 - 3.36T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 - 2.68T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 1.56T + 97T^{2} \) |
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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
−8.031373220209599685357790477351, −7.59679377123845311905991389865, −6.61893601820834221094173120816, −6.24886470852570385520367636333, −5.13077868039064078358683882620, −4.44602588341079490136554012209, −3.68222504970372363600392112628, −3.03814864175516232075485417109, −2.17938605003118437250197858413, −0.930336160147899085661994311602,
0.930336160147899085661994311602, 2.17938605003118437250197858413, 3.03814864175516232075485417109, 3.68222504970372363600392112628, 4.44602588341079490136554012209, 5.13077868039064078358683882620, 6.24886470852570385520367636333, 6.61893601820834221094173120816, 7.59679377123845311905991389865, 8.031373220209599685357790477351