| L(s) = 1 | − 2.41·2-s − 3-s + 3.82·4-s + 1.38·5-s + 2.41·6-s − 5.15·7-s − 4.41·8-s + 9-s − 3.35·10-s − 1.66·11-s − 3.82·12-s + 5.80·13-s + 12.4·14-s − 1.38·15-s + 2.99·16-s + 17-s − 2.41·18-s − 0.528·19-s + 5.31·20-s + 5.15·21-s + 4.00·22-s − 1.22·23-s + 4.41·24-s − 3.07·25-s − 14.0·26-s − 27-s − 19.7·28-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.577·3-s + 1.91·4-s + 0.621·5-s + 0.985·6-s − 1.94·7-s − 1.55·8-s + 0.333·9-s − 1.06·10-s − 0.500·11-s − 1.10·12-s + 1.61·13-s + 3.32·14-s − 0.358·15-s + 0.748·16-s + 0.242·17-s − 0.569·18-s − 0.121·19-s + 1.18·20-s + 1.12·21-s + 0.854·22-s − 0.255·23-s + 0.900·24-s − 0.614·25-s − 2.75·26-s − 0.192·27-s − 3.73·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 - 1.38T + 5T^{2} \) |
| 7 | \( 1 + 5.15T + 7T^{2} \) |
| 11 | \( 1 + 1.66T + 11T^{2} \) |
| 13 | \( 1 - 5.80T + 13T^{2} \) |
| 19 | \( 1 + 0.528T + 19T^{2} \) |
| 23 | \( 1 + 1.22T + 23T^{2} \) |
| 29 | \( 1 + 3.18T + 29T^{2} \) |
| 31 | \( 1 + 1.54T + 31T^{2} \) |
| 37 | \( 1 - 2.91T + 37T^{2} \) |
| 41 | \( 1 + 0.227T + 41T^{2} \) |
| 43 | \( 1 - 1.21T + 43T^{2} \) |
| 47 | \( 1 - 9.44T + 47T^{2} \) |
| 53 | \( 1 - 8.19T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 - 4.38T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 1.10T + 73T^{2} \) |
| 83 | \( 1 + 7.91T + 83T^{2} \) |
| 89 | \( 1 - 8.15T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−8.203738679658910341032479003823, −7.37525128212565101865791415791, −6.69199149799276480651452306762, −5.98973260166669964408937525535, −5.76114126709379043497351991921, −4.03755058989707418353522961197, −3.12950882413899508406688519037, −2.13960501525933969780704919714, −1.01566723761529106556251565720, 0,
1.01566723761529106556251565720, 2.13960501525933969780704919714, 3.12950882413899508406688519037, 4.03755058989707418353522961197, 5.76114126709379043497351991921, 5.98973260166669964408937525535, 6.69199149799276480651452306762, 7.37525128212565101865791415791, 8.203738679658910341032479003823
