| L(s) = 1 | − 2·2-s + 4·4-s − 12.7·5-s − 23.4·7-s − 8·8-s + 25.4·10-s − 11.4·13-s + 46.9·14-s + 16·16-s − 65.5·17-s + 7.25·19-s − 50.9·20-s + 104.·23-s + 37.2·25-s + 22.9·26-s − 93.8·28-s − 127.·29-s − 288.·31-s − 32·32-s + 131.·34-s + 298.·35-s − 85.4·37-s − 14.5·38-s + 101.·40-s − 135.·41-s − 353.·43-s − 208.·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.13·5-s − 1.26·7-s − 0.353·8-s + 0.805·10-s − 0.244·13-s + 0.895·14-s + 0.250·16-s − 0.934·17-s + 0.0875·19-s − 0.569·20-s + 0.943·23-s + 0.297·25-s + 0.172·26-s − 0.633·28-s − 0.815·29-s − 1.67·31-s − 0.176·32-s + 0.660·34-s + 1.44·35-s − 0.379·37-s − 0.0619·38-s + 0.402·40-s − 0.515·41-s − 1.25·43-s − 0.667·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.001514729274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001514729274\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 12.7T + 125T^{2} \) |
| 7 | \( 1 + 23.4T + 343T^{2} \) |
| 13 | \( 1 + 11.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 65.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 7.25T + 6.85e3T^{2} \) |
| 23 | \( 1 - 104.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 127.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 288.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 85.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 135.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 353.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 134.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 501.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 651.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 365.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 294.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 132.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 469.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 408.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.35e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 260.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.41e3T + 9.12e5T^{2} \) |
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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
−8.892197689037079370545314536990, −7.895039681938342102937315801640, −7.20399320347973970141502857771, −6.69409491912968976496745075576, −5.72566274836905835965786195785, −4.57808179033098912778624789979, −3.56711268335696752632562461342, −2.98403378878724360235660966097, −1.65952322311041033857719966128, −0.01899076508678294411270597476,
0.01899076508678294411270597476, 1.65952322311041033857719966128, 2.98403378878724360235660966097, 3.56711268335696752632562461342, 4.57808179033098912778624789979, 5.72566274836905835965786195785, 6.69409491912968976496745075576, 7.20399320347973970141502857771, 7.895039681938342102937315801640, 8.892197689037079370545314536990
