| L(s) = 1 | + 2-s + 4-s + 5-s + 4·7-s + 8-s + 10-s − 6·11-s + 13-s + 4·14-s + 16-s + 2·19-s + 20-s − 6·22-s + 6·23-s + 25-s + 26-s + 4·28-s − 6·29-s − 2·31-s + 32-s + 4·35-s − 2·37-s + 2·38-s + 40-s − 6·41-s + 2·43-s − 6·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.51·7-s + 0.353·8-s + 0.316·10-s − 1.80·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.458·19-s + 0.223·20-s − 1.27·22-s + 1.25·23-s + 1/5·25-s + 0.196·26-s + 0.755·28-s − 1.11·29-s − 0.359·31-s + 0.176·32-s + 0.676·35-s − 0.328·37-s + 0.324·38-s + 0.158·40-s − 0.937·41-s + 0.304·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−12.78598066617978, −12.46423639459305, −11.97279084736204, −11.28690885442664, −11.01900659259398, −10.74795550255835, −10.28843098112054, −9.709196006246884, −9.025488905287737, −8.708758607744120, −8.047871401057640, −7.670647830465347, −7.353913004258960, −6.854518203607514, −6.006742145897008, −5.664641255723016, −5.193135377260704, −4.889372604849002, −4.515226198541705, −3.679273957063453, −3.217139623935709, −2.541156030626912, −2.169240416567661, −1.547402035249514, −0.9955197770085586, 0,
0.9955197770085586, 1.547402035249514, 2.169240416567661, 2.541156030626912, 3.217139623935709, 3.679273957063453, 4.515226198541705, 4.889372604849002, 5.193135377260704, 5.664641255723016, 6.006742145897008, 6.854518203607514, 7.353913004258960, 7.670647830465347, 8.047871401057640, 8.708758607744120, 9.025488905287737, 9.709196006246884, 10.28843098112054, 10.74795550255835, 11.01900659259398, 11.28690885442664, 11.97279084736204, 12.46423639459305, 12.78598066617978
