L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 4·7-s + 8-s + 9-s + 11-s − 12-s − 2·13-s + 4·14-s + 16-s + 17-s + 18-s − 4·19-s − 4·21-s + 22-s − 24-s − 2·26-s − 27-s + 4·28-s − 6·29-s + 8·31-s + 32-s − 33-s + 34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s − 0.872·21-s + 0.213·22-s − 0.204·24-s − 0.392·26-s − 0.192·27-s + 0.755·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.174·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.872055951\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.872055951\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.87187306248480, −14.69523851461021, −14.45885189972758, −13.48397956725990, −13.17973821129304, −12.49834551003345, −11.89621337549599, −11.60926258796293, −10.95224913986872, −10.74073184482449, −9.839754161617794, −9.393038394377833, −8.410792158417748, −8.007935584734324, −7.458381900350949, −6.796442478843928, −6.102647955698118, −5.650273888429101, −4.897273035070660, −4.480998267033680, −4.068955150515871, −3.014240870027720, −2.212817396938299, −1.601460391408026, −0.7213912088932808,
0.7213912088932808, 1.601460391408026, 2.212817396938299, 3.014240870027720, 4.068955150515871, 4.480998267033680, 4.897273035070660, 5.650273888429101, 6.102647955698118, 6.796442478843928, 7.458381900350949, 8.007935584734324, 8.410792158417748, 9.393038394377833, 9.839754161617794, 10.74073184482449, 10.95224913986872, 11.60926258796293, 11.89621337549599, 12.49834551003345, 13.17973821129304, 13.48397956725990, 14.45885189972758, 14.69523851461021, 14.87187306248480