L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 6.10·11-s + 12-s + 1.68·13-s + 16-s − 6.83·17-s − 18-s − 1.68·19-s − 6.10·22-s − 2.31·23-s − 24-s − 1.68·26-s + 27-s + 8.41·29-s + 0.688·31-s − 32-s + 6.10·33-s + 6.83·34-s + 36-s − 2.31·37-s + 1.68·38-s + 1.68·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 1.84·11-s + 0.288·12-s + 0.468·13-s + 0.250·16-s − 1.65·17-s − 0.235·18-s − 0.387·19-s − 1.30·22-s − 0.481·23-s − 0.204·24-s − 0.331·26-s + 0.192·27-s + 1.56·29-s + 0.123·31-s − 0.176·32-s + 1.06·33-s + 1.17·34-s + 0.166·36-s − 0.380·37-s + 0.273·38-s + 0.270·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.034607874\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.034607874\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 6.10T + 11T^{2} \) |
| 13 | \( 1 - 1.68T + 13T^{2} \) |
| 17 | \( 1 + 6.83T + 17T^{2} \) |
| 19 | \( 1 + 1.68T + 19T^{2} \) |
| 23 | \( 1 + 2.31T + 23T^{2} \) |
| 29 | \( 1 - 8.41T + 29T^{2} \) |
| 31 | \( 1 - 0.688T + 31T^{2} \) |
| 37 | \( 1 + 2.31T + 37T^{2} \) |
| 41 | \( 1 + 5.14T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 3.06T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 7.04T + 59T^{2} \) |
| 61 | \( 1 + 9.46T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 7.31T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 4.95T + 97T^{2} \) |
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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
−8.239848833580137693316170282098, −7.15404297220493476945898309969, −6.49576193403475166494614575567, −6.35259122499569329059333493987, −4.97875712689319574788622976057, −4.10771591859139313711572086216, −3.60383971405804925623047761891, −2.47232091568639919941227185873, −1.77124229290095599132467697911, −0.800816457618215052854058111327,
0.800816457618215052854058111327, 1.77124229290095599132467697911, 2.47232091568639919941227185873, 3.60383971405804925623047761891, 4.10771591859139313711572086216, 4.97875712689319574788622976057, 6.35259122499569329059333493987, 6.49576193403475166494614575567, 7.15404297220493476945898309969, 8.239848833580137693316170282098