| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 11-s + 3·13-s − 14-s + 16-s + 7·17-s − 6·19-s − 22-s − 3·23-s − 3·26-s + 28-s − 3·29-s − 7·31-s − 32-s − 7·34-s − 2·37-s + 6·38-s + 8·41-s + 5·43-s + 44-s + 3·46-s + 47-s + 49-s + 3·52-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 + 1.69·17-s − 1.37·19-s − 0.213·22-s − 0.625·23-s − 0.588·26-s + 0.188·28-s − 0.557·29-s − 1.25·31-s − 0.176·32-s − 1.20·34-s − 0.328·37-s + 0.973·38-s + 1.24·41-s + 0.762·43-s + 0.150·44-s + 0.442·46-s + 0.145·47-s + 1/7·49-s + 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.611882688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.611882688\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.72184191924715745056140510123, −7.26826171759744277021789902414, −6.24736215579196282124139594229, −5.88826626492015426104897336871, −5.06258223033648556501084118987, −3.96669496654903169934356866679, −3.53837786526680846495879854310, −2.36544926881313302453573122414, −1.62708759035717561481706779252, −0.70176961551650889998448025675,
0.70176961551650889998448025675, 1.62708759035717561481706779252, 2.36544926881313302453573122414, 3.53837786526680846495879854310, 3.96669496654903169934356866679, 5.06258223033648556501084118987, 5.88826626492015426104897336871, 6.24736215579196282124139594229, 7.26826171759744277021789902414, 7.72184191924715745056140510123
