| L(s) = 1 | − 17.1·3-s − 52.0·5-s − 180.·7-s + 50.3·9-s + 401.·11-s − 591.·13-s + 891.·15-s − 39.9·17-s − 1.67e3·19-s + 3.09e3·21-s + 1.06e3·23-s − 416.·25-s + 3.30e3·27-s − 6.12e3·29-s − 6.67e3·31-s − 6.87e3·33-s + 9.40e3·35-s + 8.96e3·37-s + 1.01e4·39-s + 1.90e4·41-s + 1.36e4·43-s − 2.61e3·45-s − 6.95e3·47-s + 1.58e4·49-s + 685.·51-s + 2.16e4·53-s − 2.08e4·55-s + ⋯ |
| L(s) = 1 | − 1.09·3-s − 0.930·5-s − 1.39·7-s + 0.207·9-s + 0.999·11-s − 0.970·13-s + 1.02·15-s − 0.0335·17-s − 1.06·19-s + 1.53·21-s + 0.417·23-s − 0.133·25-s + 0.871·27-s − 1.35·29-s − 1.24·31-s − 1.09·33-s + 1.29·35-s + 1.07·37-s + 1.06·39-s + 1.76·41-s + 1.12·43-s − 0.192·45-s − 0.459·47-s + 0.942·49-s + 0.0368·51-s + 1.06·53-s − 0.930·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 + 17.1T + 243T^{2} \) |
| 5 | \( 1 + 52.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + 180.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 401.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 591.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 39.9T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.67e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.06e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.12e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.67e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.96e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.90e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.36e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.95e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.16e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.72e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.00e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.72e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.86e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.08e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.30e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.06e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.48e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.56e4T + 8.58e9T^{2} \) |
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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
−9.048892758688114948767815492581, −7.72818489557274296125127855747, −6.96342607937605648306497440188, −6.25903957140085334130774127545, −5.51534093306368471683847078561, −4.29251961595557285069276952986, −3.66203800069184565551769078389, −2.40937060822544038070410136127, −0.69880570014987991668594958136, 0,
0.69880570014987991668594958136, 2.40937060822544038070410136127, 3.66203800069184565551769078389, 4.29251961595557285069276952986, 5.51534093306368471683847078561, 6.25903957140085334130774127545, 6.96342607937605648306497440188, 7.72818489557274296125127855747, 9.048892758688114948767815492581
