| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s − 11-s − 2·13-s − 4·14-s + 16-s − 17-s − 4·19-s + 22-s + 2·26-s + 4·28-s + 6·29-s + 8·31-s − 32-s + 34-s + 10·37-s + 4·38-s − 6·41-s + 4·43-s − 44-s + 12·47-s + 9·49-s − 2·52-s + 6·53-s − 4·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 0.301·11-s − 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.917·19-s + 0.213·22-s + 0.392·26-s + 0.755·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.171·34-s + 1.64·37-s + 0.648·38-s − 0.937·41-s + 0.609·43-s − 0.150·44-s + 1.75·47-s + 9/7·49-s − 0.277·52-s + 0.824·53-s − 0.534·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.210467645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.210467645\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 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
−13.88421872751830, −13.67214185916157, −12.83259040468581, −12.25423456280380, −11.94100750475339, −11.35975027790884, −10.90273121792152, −10.46742383806298, −10.04885143408706, −9.376351859905856, −8.802082005420109, −8.333132821826879, −7.972252821042408, −7.510078858300951, −6.892908663825388, −6.299953065505070, −5.732950563630516, −5.049489743276838, −4.438586598182164, −4.232475601269449, −3.042730560007707, −2.463289660274329, −2.023809275248977, −1.182048488976062, −0.5893711886728673,
0.5893711886728673, 1.182048488976062, 2.023809275248977, 2.463289660274329, 3.042730560007707, 4.232475601269449, 4.438586598182164, 5.049489743276838, 5.732950563630516, 6.299953065505070, 6.892908663825388, 7.510078858300951, 7.972252821042408, 8.333132821826879, 8.802082005420109, 9.376351859905856, 10.04885143408706, 10.46742383806298, 10.90273121792152, 11.35975027790884, 11.94100750475339, 12.25423456280380, 12.83259040468581, 13.67214185916157, 13.88421872751830
