| L(s) = 1 | + 2-s + 4-s − 2.44·5-s + 4.44·7-s + 8-s − 2.44·10-s + 4.89·11-s − 2.89·13-s + 4.44·14-s + 16-s − 17-s + 6.89·19-s − 2.44·20-s + 4.89·22-s − 8.44·23-s + 0.999·25-s − 2.89·26-s + 4.44·28-s + 2.44·29-s − 5.34·31-s + 32-s − 34-s − 10.8·35-s − 6.44·37-s + 6.89·38-s − 2.44·40-s − 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.09·5-s + 1.68·7-s + 0.353·8-s − 0.774·10-s + 1.47·11-s − 0.804·13-s + 1.18·14-s + 0.250·16-s − 0.242·17-s + 1.58·19-s − 0.547·20-s + 1.04·22-s − 1.76·23-s + 0.199·25-s − 0.568·26-s + 0.840·28-s + 0.454·29-s − 0.960·31-s + 0.176·32-s − 0.171·34-s − 1.84·35-s − 1.06·37-s + 1.11·38-s − 0.387·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.961596382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.961596382\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2.44T + 5T^{2} \) |
| 7 | \( 1 - 4.44T + 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 19 | \( 1 - 6.89T + 19T^{2} \) |
| 23 | \( 1 + 8.44T + 23T^{2} \) |
| 29 | \( 1 - 2.44T + 29T^{2} \) |
| 31 | \( 1 + 5.34T + 31T^{2} \) |
| 37 | \( 1 + 6.44T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 4.89T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 1.10T + 59T^{2} \) |
| 61 | \( 1 + 1.55T + 61T^{2} \) |
| 67 | \( 1 - 6.89T + 67T^{2} \) |
| 71 | \( 1 - 1.34T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 5.34T + 79T^{2} \) |
| 83 | \( 1 - 1.10T + 83T^{2} \) |
| 89 | \( 1 + 3.79T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.84113585983832538063280761474, −11.30194701010089363838751784007, −9.992046172226905601459623132759, −8.613410583582073825287345107565, −7.74357543422960419268710618884, −6.98757685114717043892086454957, −5.45364476254799833238769270848, −4.47768029060519589093036931803, −3.64098565511974959530421417733, −1.71772058640558074185347431055,
1.71772058640558074185347431055, 3.64098565511974959530421417733, 4.47768029060519589093036931803, 5.45364476254799833238769270848, 6.98757685114717043892086454957, 7.74357543422960419268710618884, 8.613410583582073825287345107565, 9.992046172226905601459623132759, 11.30194701010089363838751784007, 11.84113585983832538063280761474
