L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 3·7-s + 8-s + 9-s + 3·11-s + 12-s + 2·13-s + 3·14-s + 16-s + 4·17-s + 18-s − 3·19-s + 3·21-s + 3·22-s − 5·23-s + 24-s + 2·26-s + 27-s + 3·28-s + 4·29-s + 31-s + 32-s + 3·33-s + 4·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s + 0.554·13-s + 0.801·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.688·19-s + 0.654·21-s + 0.639·22-s − 1.04·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.566·28-s + 0.742·29-s + 0.179·31-s + 0.176·32-s + 0.522·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.043265576\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.043265576\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.179465993255970405884074967119, −7.74153241357675715017471088373, −6.74551289508181603008919709039, −6.13324876690007371037739885009, −5.26698158762698358805063469928, −4.42776683955685438460685738324, −3.89612651996431308605905760703, −3.01714077393479385545797310097, −1.94548506923537729035949511135, −1.24928453466983135859884605139,
1.24928453466983135859884605139, 1.94548506923537729035949511135, 3.01714077393479385545797310097, 3.89612651996431308605905760703, 4.42776683955685438460685738324, 5.26698158762698358805063469928, 6.13324876690007371037739885009, 6.74551289508181603008919709039, 7.74153241357675715017471088373, 8.179465993255970405884074967119