| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 5·11-s + 12-s − 3·13-s + 15-s + 16-s + 2·17-s + 18-s − 5·19-s + 20-s + 5·22-s − 3·23-s + 24-s + 25-s − 3·26-s + 27-s − 6·29-s + 30-s + 4·31-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 + 1.50·11-s + 0.288·12-s − 0.832·13-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 1.14·19-s + 0.223·20-s + 1.06·22-s − 0.625·23-s + 0.204·24-s + 1/5·25-s − 0.588·26-s + 0.192·27-s − 1.11·29-s + 0.182·30-s + 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.58943853898075, −13.95644170952440, −13.60075881124527, −13.14345424293790, −12.42382697243903, −12.10082700962078, −11.73418268878076, −10.94757887715855, −10.47241201851525, −9.794075178828699, −9.519258274280030, −8.875947553218448, −8.302036273983050, −7.792839500397664, −6.946652421066312, −6.770662574325429, −6.079514112471280, −5.572255982934104, −4.787933550869815, −4.327749880799393, −3.721280833075107, −3.207439722835582, −2.427758710976185, −1.810723387862924, −1.332272674070599, 0,
1.332272674070599, 1.810723387862924, 2.427758710976185, 3.207439722835582, 3.721280833075107, 4.327749880799393, 4.787933550869815, 5.572255982934104, 6.079514112471280, 6.770662574325429, 6.946652421066312, 7.792839500397664, 8.302036273983050, 8.875947553218448, 9.519258274280030, 9.794075178828699, 10.47241201851525, 10.94757887715855, 11.73418268878076, 12.10082700962078, 12.42382697243903, 13.14345424293790, 13.60075881124527, 13.95644170952440, 14.58943853898075
