| L(s) = 1 | + 5.41·2-s + 4.65·3-s + 21.3·4-s + 5·5-s + 25.2·6-s + 72.0·8-s − 5.31·9-s + 27.0·10-s − 52.2·11-s + 99.2·12-s − 30.6·13-s + 23.2·15-s + 219.·16-s − 37.2·17-s − 28.7·18-s − 80.2·19-s + 106.·20-s − 282.·22-s + 25.8·23-s + 335.·24-s + 25·25-s − 165.·26-s − 150.·27-s + 20.9·29-s + 126.·30-s + 314.·31-s + 613.·32-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 0.896·3-s + 2.66·4-s + 0.447·5-s + 1.71·6-s + 3.18·8-s − 0.196·9-s + 0.856·10-s − 1.43·11-s + 2.38·12-s − 0.654·13-s + 0.400·15-s + 3.43·16-s − 0.531·17-s − 0.376·18-s − 0.968·19-s + 1.19·20-s − 2.74·22-s + 0.234·23-s + 2.85·24-s + 0.200·25-s − 1.25·26-s − 1.07·27-s + 0.134·29-s + 0.767·30-s + 1.82·31-s + 3.38·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.162652047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.162652047\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 5.41T + 8T^{2} \) |
| 3 | \( 1 - 4.65T + 27T^{2} \) |
| 11 | \( 1 + 52.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 30.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 37.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 80.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 25.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 20.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 314.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 197.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 11.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 33.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 361.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 153.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 616T + 2.05e5T^{2} \) |
| 61 | \( 1 + 15.2T + 2.26e5T^{2} \) |
| 67 | \( 1 + 166.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 952T + 3.57e5T^{2} \) |
| 73 | \( 1 - 148.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 857.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 660.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 45.7T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.68e3T + 9.12e5T^{2} \) |
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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
−12.00744182014384802302848586624, −10.93307855172238321430650419227, −10.05158467906896623283892956210, −8.439499799763868921683104019687, −7.43367384218398661833196368313, −6.26931502647373891962714963613, −5.26250989570475182922531702127, −4.27316770750334373277901485334, −2.79619774893889984447741283676, −2.35078180802157557984254140315,
2.35078180802157557984254140315, 2.79619774893889984447741283676, 4.27316770750334373277901485334, 5.26250989570475182922531702127, 6.26931502647373891962714963613, 7.43367384218398661833196368313, 8.439499799763868921683104019687, 10.05158467906896623283892956210, 10.93307855172238321430650419227, 12.00744182014384802302848586624
