L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 6·11-s + 12-s + 13-s + 15-s + 16-s − 17-s − 18-s − 19-s + 20-s − 6·22-s − 4·23-s − 24-s + 25-s − 26-s + 27-s − 8·29-s − 30-s + 3·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.80·11-s + 0.288·12-s + 0.277·13-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s − 1.27·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 1.48·29-s − 0.182·30-s + 0.538·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.67535042674307, −12.39477215064819, −11.90469505421089, −11.41730650198783, −10.94185858925092, −10.57115510293470, −9.806061592734426, −9.689690130272435, −9.052411399986013, −8.912741038979559, −8.375570203494612, −7.805391310756726, −7.375327962934324, −6.786062700470244, −6.397702265920737, −6.058026675959301, −5.407274989774164, −4.746514351564686, −4.046971304242273, −3.704585027033063, −3.268261804301027, −2.311854619400424, −2.062483576422498, −1.419452569091133, −0.9419478967457949, 0,
0.9419478967457949, 1.419452569091133, 2.062483576422498, 2.311854619400424, 3.268261804301027, 3.704585027033063, 4.046971304242273, 4.746514351564686, 5.407274989774164, 6.058026675959301, 6.397702265920737, 6.786062700470244, 7.375327962934324, 7.805391310756726, 8.375570203494612, 8.912741038979559, 9.052411399986013, 9.689690130272435, 9.806061592734426, 10.57115510293470, 10.94185858925092, 11.41730650198783, 11.90469505421089, 12.39477215064819, 12.67535042674307