L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 0.554·7-s + 8-s + 9-s + 10-s + 1.35·11-s + 12-s + 0.554·14-s + 15-s + 16-s + 1.80·17-s + 18-s + 3.33·19-s + 20-s + 0.554·21-s + 1.35·22-s − 0.692·23-s + 24-s + 25-s + 27-s + 0.554·28-s + 5.04·29-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 + 0.209·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.409·11-s + 0.288·12-s + 0.148·14-s + 0.258·15-s + 0.250·16-s + 0.437·17-s + 0.235·18-s + 0.765·19-s + 0.223·20-s + 0.121·21-s + 0.289·22-s − 0.144·23-s + 0.204·24-s + 0.200·25-s + 0.192·27-s + 0.104·28-s + 0.937·29-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
\(L(1)\) |
\(\approx\) |
\(4.878282559\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.878282559\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.554T + 7T^{2} \) |
| 11 | \( 1 - 1.35T + 11T^{2} \) |
| 17 | \( 1 - 1.80T + 17T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 + 0.692T + 23T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 + 0.158T + 37T^{2} \) |
| 41 | \( 1 + 0.664T + 41T^{2} \) |
| 43 | \( 1 + 3.96T + 43T^{2} \) |
| 47 | \( 1 + 2.10T + 47T^{2} \) |
| 53 | \( 1 - 1.15T + 53T^{2} \) |
| 59 | \( 1 + 8.72T + 59T^{2} \) |
| 61 | \( 1 + 5.56T + 61T^{2} \) |
| 67 | \( 1 - 3.58T + 67T^{2} \) |
| 71 | \( 1 - 4.96T + 71T^{2} \) |
| 73 | \( 1 - 3.70T + 73T^{2} \) |
| 79 | \( 1 - 3.25T + 79T^{2} \) |
| 83 | \( 1 - 3.85T + 83T^{2} \) |
| 89 | \( 1 - 3.13T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−8.114163937146314075337360058720, −7.52938823359923448771142319874, −6.65280093414687398219651639186, −6.11535275690833460240595791662, −5.15378351949428935612638437411, −4.61837866523055272149449769588, −3.60323443545946864132829253048, −3.00919781968033929000559997517, −2.04580538010526786234632862404, −1.14906591079811226552159319459,
1.14906591079811226552159319459, 2.04580538010526786234632862404, 3.00919781968033929000559997517, 3.60323443545946864132829253048, 4.61837866523055272149449769588, 5.15378351949428935612638437411, 6.11535275690833460240595791662, 6.65280093414687398219651639186, 7.52938823359923448771142319874, 8.114163937146314075337360058720