| L(s) = 1 | + 2-s + 4-s − 4.23·5-s + 7-s + 8-s − 4.23·10-s − 1.38·13-s + 14-s + 16-s + 1.61·17-s + 3.85·19-s − 4.23·20-s − 6.23·23-s + 12.9·25-s − 1.38·26-s + 28-s + 0.381·29-s + 8.70·31-s + 32-s + 1.61·34-s − 4.23·35-s − 4.85·37-s + 3.85·38-s − 4.23·40-s − 10.4·41-s − 10.8·43-s − 6.23·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.89·5-s + 0.377·7-s + 0.353·8-s − 1.33·10-s − 0.383·13-s + 0.267·14-s + 0.250·16-s + 0.392·17-s + 0.884·19-s − 0.947·20-s − 1.30·23-s + 2.58·25-s − 0.271·26-s + 0.188·28-s + 0.0709·29-s + 1.56·31-s + 0.176·32-s + 0.277·34-s − 0.716·35-s − 0.798·37-s + 0.625·38-s − 0.669·40-s − 1.63·41-s − 1.65·43-s − 0.919·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 4.23T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 13 | \( 1 + 1.38T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 - 3.85T + 19T^{2} \) |
| 23 | \( 1 + 6.23T + 23T^{2} \) |
| 29 | \( 1 - 0.381T + 29T^{2} \) |
| 31 | \( 1 - 8.70T + 31T^{2} \) |
| 37 | \( 1 + 4.85T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 0.527T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 2.61T + 59T^{2} \) |
| 61 | \( 1 - 7.61T + 61T^{2} \) |
| 67 | \( 1 - 7.76T + 67T^{2} \) |
| 71 | \( 1 + 5.61T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 5.23T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 7.14T + 97T^{2} \) |
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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
−7.60026831073586241850835244658, −7.04159471360341451010263059351, −6.33870246046032168762603622790, −5.13757419649980731450193914863, −4.84644925712119343615617406336, −3.86617796801316403902113042968, −3.50789712184087602981996274937, −2.58896387050594435837991240588, −1.28390625679216484733065236253, 0,
1.28390625679216484733065236253, 2.58896387050594435837991240588, 3.50789712184087602981996274937, 3.86617796801316403902113042968, 4.84644925712119343615617406336, 5.13757419649980731450193914863, 6.33870246046032168762603622790, 7.04159471360341451010263059351, 7.60026831073586241850835244658
