| L(s) = 1 | − 9·3-s + 48.9·5-s + 70.3·7-s + 81·9-s + 461.·11-s − 157.·13-s − 440.·15-s + 225.·17-s − 1.51e3·19-s − 632.·21-s − 529·23-s − 729.·25-s − 729·27-s + 312.·29-s + 6.61e3·31-s − 4.15e3·33-s + 3.44e3·35-s − 9.13e3·37-s + 1.42e3·39-s − 1.74e4·41-s − 4.41e3·43-s + 3.96e3·45-s − 2.87e4·47-s − 1.18e4·49-s − 2.02e3·51-s − 3.74e4·53-s + 2.26e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.875·5-s + 0.542·7-s + 0.333·9-s + 1.15·11-s − 0.259·13-s − 0.505·15-s + 0.189·17-s − 0.964·19-s − 0.313·21-s − 0.208·23-s − 0.233·25-s − 0.192·27-s + 0.0690·29-s + 1.23·31-s − 0.664·33-s + 0.474·35-s − 1.09·37-s + 0.149·39-s − 1.62·41-s − 0.363·43-s + 0.291·45-s − 1.89·47-s − 0.705·49-s − 0.109·51-s − 1.83·53-s + 1.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 + 529T \) |
| good | 5 | \( 1 - 48.9T + 3.12e3T^{2} \) |
| 7 | \( 1 - 70.3T + 1.68e4T^{2} \) |
| 11 | \( 1 - 461.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 157.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 225.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.51e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 312.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.61e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.13e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.74e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.41e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.87e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.74e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.45e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.81e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.49e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.07e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.71e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.63e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.26e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.97e4T + 8.58e9T^{2} \) |
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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
−8.731194345055428785898113996903, −7.968161645907920904731354405676, −6.62884234819005727965876624159, −6.38862342258592129902228594563, −5.25297444711385945926594650002, −4.57172467623596361230951121844, −3.43946224629187010310182253181, −1.98907443092128395826139632250, −1.37536512109689944990901846274, 0,
1.37536512109689944990901846274, 1.98907443092128395826139632250, 3.43946224629187010310182253181, 4.57172467623596361230951121844, 5.25297444711385945926594650002, 6.38862342258592129902228594563, 6.62884234819005727965876624159, 7.968161645907920904731354405676, 8.731194345055428785898113996903
