| L(s) = 1 | − 1.42·2-s − 29.9·4-s + 83.8·5-s + 175.·7-s + 88.5·8-s − 119.·10-s − 373.·11-s + 930.·13-s − 250.·14-s + 832.·16-s + 1.10e3·17-s − 2.42e3·19-s − 2.51e3·20-s + 533.·22-s − 529·23-s + 3.91e3·25-s − 1.33e3·26-s − 5.26e3·28-s − 7.81e3·29-s + 2.72e3·31-s − 4.02e3·32-s − 1.58e3·34-s + 1.47e4·35-s + 1.28e4·37-s + 3.47e3·38-s + 7.42e3·40-s + 1.22e4·41-s + ⋯ |
| L(s) = 1 | − 0.252·2-s − 0.936·4-s + 1.50·5-s + 1.35·7-s + 0.489·8-s − 0.379·10-s − 0.930·11-s + 1.52·13-s − 0.342·14-s + 0.812·16-s + 0.930·17-s − 1.54·19-s − 1.40·20-s + 0.234·22-s − 0.208·23-s + 1.25·25-s − 0.385·26-s − 1.26·28-s − 1.72·29-s + 0.509·31-s − 0.694·32-s − 0.235·34-s + 2.03·35-s + 1.54·37-s + 0.389·38-s + 0.733·40-s + 1.13·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.260210887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.260210887\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 + 1.42T + 32T^{2} \) |
| 5 | \( 1 - 83.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 175.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 373.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 930.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.10e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.42e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 7.81e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.72e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.28e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.22e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.26e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.48e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.38e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.00e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.52e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.98e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.34e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.00e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.95e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.44e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.24e4T + 8.58e9T^{2} \) |
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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
−11.17792764912814661090121108517, −10.47058160711021132566970494889, −9.533437058438504417178049966757, −8.541335196901399637921608729176, −7.85696891356086880467364636023, −5.99187099855813616953810507633, −5.33096213755705846804312019725, −4.11243383802486836108475685957, −2.14224481552424004784350205198, −1.04991683693279752872352327791,
1.04991683693279752872352327791, 2.14224481552424004784350205198, 4.11243383802486836108475685957, 5.33096213755705846804312019725, 5.99187099855813616953810507633, 7.85696891356086880467364636023, 8.541335196901399637921608729176, 9.533437058438504417178049966757, 10.47058160711021132566970494889, 11.17792764912814661090121108517
