| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 11-s + 3·13-s + 14-s + 16-s + 4·17-s + 22-s − 7·23-s − 3·26-s − 28-s + 4·29-s + 2·31-s − 32-s − 4·34-s − 8·37-s + 10·41-s + 11·43-s − 44-s + 7·46-s − 47-s − 6·49-s + 3·52-s + 56-s − 4·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.301·11-s + 0.832·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.213·22-s − 1.45·23-s − 0.588·26-s − 0.188·28-s + 0.742·29-s + 0.359·31-s − 0.176·32-s − 0.685·34-s − 1.31·37-s + 1.56·41-s + 1.67·43-s − 0.150·44-s + 1.03·46-s − 0.145·47-s − 6/7·49-s + 0.416·52-s + 0.133·56-s − 0.525·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.850005163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.850005163\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−12.85443343911214, −12.32803035602200, −12.11051879016081, −11.45929158777955, −11.00720603642346, −10.55521727507476, −9.998243662018359, −9.858535281531452, −9.157518432386026, −8.729979586244449, −8.227773309788253, −7.790786265135715, −7.422932867844057, −6.745716637745373, −6.218498116562410, −5.903089988335120, −5.366173187612188, −4.681630732102609, −3.950377826053261, −3.587638288090297, −2.931136771316972, −2.359409215481088, −1.750895859215165, −0.9962957587946585, −0.4964605727079357,
0.4964605727079357, 0.9962957587946585, 1.750895859215165, 2.359409215481088, 2.931136771316972, 3.587638288090297, 3.950377826053261, 4.681630732102609, 5.366173187612188, 5.903089988335120, 6.218498116562410, 6.745716637745373, 7.422932867844057, 7.790786265135715, 8.227773309788253, 8.729979586244449, 9.157518432386026, 9.858535281531452, 9.998243662018359, 10.55521727507476, 11.00720603642346, 11.45929158777955, 12.11051879016081, 12.32803035602200, 12.85443343911214
