| L(s) = 1 | − 2-s + 4-s − 3.69·5-s − 2.24·7-s − 8-s + 3.69·10-s + 5.56·13-s + 2.24·14-s + 16-s + 1.01·17-s − 2.65·19-s − 3.69·20-s − 2.65·23-s + 8.62·25-s − 5.56·26-s − 2.24·28-s + 1.38·29-s + 4.89·31-s − 32-s − 1.01·34-s + 8.27·35-s − 1.97·37-s + 2.65·38-s + 3.69·40-s − 4.28·41-s + 0.551·43-s + 2.65·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.65·5-s − 0.847·7-s − 0.353·8-s + 1.16·10-s + 1.54·13-s + 0.599·14-s + 0.250·16-s + 0.246·17-s − 0.608·19-s − 0.825·20-s − 0.552·23-s + 1.72·25-s − 1.09·26-s − 0.423·28-s + 0.256·29-s + 0.878·31-s − 0.176·32-s − 0.174·34-s + 1.39·35-s − 0.325·37-s + 0.430·38-s + 0.583·40-s − 0.669·41-s + 0.0840·43-s + 0.390·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 3.69T + 5T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 5.56T + 13T^{2} \) |
| 17 | \( 1 - 1.01T + 17T^{2} \) |
| 19 | \( 1 + 2.65T + 19T^{2} \) |
| 23 | \( 1 + 2.65T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 + 1.97T + 37T^{2} \) |
| 41 | \( 1 + 4.28T + 41T^{2} \) |
| 43 | \( 1 - 0.551T + 43T^{2} \) |
| 47 | \( 1 - 5.62T + 47T^{2} \) |
| 53 | \( 1 + 1.03T + 53T^{2} \) |
| 59 | \( 1 - 2.42T + 59T^{2} \) |
| 61 | \( 1 - 5.38T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 - 8.68T + 71T^{2} \) |
| 73 | \( 1 + 1.75T + 73T^{2} \) |
| 79 | \( 1 - 3.74T + 79T^{2} \) |
| 83 | \( 1 - 5.43T + 83T^{2} \) |
| 89 | \( 1 - 6.27T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
| 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
−8.150536068773295054467727792094, −8.005966228574809576067323212192, −6.74032547757760489969429421109, −6.53683481708450856963098024370, −5.35065253997113462685789840248, −4.01118482076769723664379969829, −3.68757852859161312793919658963, −2.68099815052239903953087550753, −1.11466830082390253149866671954, 0,
1.11466830082390253149866671954, 2.68099815052239903953087550753, 3.68757852859161312793919658963, 4.01118482076769723664379969829, 5.35065253997113462685789840248, 6.53683481708450856963098024370, 6.74032547757760489969429421109, 8.005966228574809576067323212192, 8.150536068773295054467727792094
