L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 7-s − 8-s + 9-s − 2·10-s + 3·11-s − 12-s + 6·13-s + 14-s − 2·15-s + 16-s + 6·17-s − 18-s + 2·20-s + 21-s − 3·22-s + 23-s + 24-s − 25-s − 6·26-s − 27-s − 28-s + 2·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.904·11-s − 0.288·12-s + 1.66·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.447·20-s + 0.218·21-s − 0.639·22-s + 0.208·23-s + 0.204·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.188·28-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.639167401\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.639167401\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−12.43858507183493, −11.91882877078566, −11.62478718825175, −11.24108407603428, −10.55381583992897, −10.25744932011241, −9.790276246329706, −9.574288953252393, −8.897731171148926, −8.508793512628891, −8.079134910453932, −7.511177314337070, −6.786365337807233, −6.506181425253967, −6.045432840003630, −5.865008031179678, −5.081593908244060, −4.698419028007575, −3.756674481631849, −3.397015473261235, −3.029547793566813, −1.917827096079351, −1.655891719036653, −1.072319508156785, −0.5579481328107797,
0.5579481328107797, 1.072319508156785, 1.655891719036653, 1.917827096079351, 3.029547793566813, 3.397015473261235, 3.756674481631849, 4.698419028007575, 5.081593908244060, 5.865008031179678, 6.045432840003630, 6.506181425253967, 6.786365337807233, 7.511177314337070, 8.079134910453932, 8.508793512628891, 8.897731171148926, 9.574288953252393, 9.790276246329706, 10.25744932011241, 10.55381583992897, 11.24108407603428, 11.62478718825175, 11.91882877078566, 12.43858507183493