| L(s) = 1 | − 9·3-s − 46.2·5-s − 245.·7-s + 81·9-s − 711.·11-s + 539.·13-s + 415.·15-s + 433.·17-s − 471.·19-s + 2.20e3·21-s − 529·23-s − 988.·25-s − 729·27-s + 6.05e3·29-s + 4.31e3·31-s + 6.40e3·33-s + 1.13e4·35-s − 1.28e3·37-s − 4.85e3·39-s − 1.28e4·41-s + 7.25e3·43-s − 3.74e3·45-s + 2.35e4·47-s + 4.32e4·49-s − 3.89e3·51-s + 9.91e3·53-s + 3.28e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.826·5-s − 1.89·7-s + 0.333·9-s − 1.77·11-s + 0.885·13-s + 0.477·15-s + 0.363·17-s − 0.299·19-s + 1.09·21-s − 0.208·23-s − 0.316·25-s − 0.192·27-s + 1.33·29-s + 0.805·31-s + 1.02·33-s + 1.56·35-s − 0.154·37-s − 0.511·39-s − 1.18·41-s + 0.598·43-s − 0.275·45-s + 1.55·47-s + 2.57·49-s − 0.209·51-s + 0.484·53-s + 1.46·55-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 | \( 1 + 46.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 245.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 711.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 539.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 433.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 471.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 6.05e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.31e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.28e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.28e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.25e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.35e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.91e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 9.91e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.32e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.33e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.96e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.36e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.72e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.70e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.40e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.84e4T + 8.58e9T^{2} \) |
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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
−8.623076519855784723914258995466, −7.85124170632818152646690634321, −6.95313247714358247925338231094, −6.18409633588450100739691831283, −5.44823180660550178627592695111, −4.26365735367184896287867240777, −3.37202405059832626714624737197, −2.58149437427144896698579904662, −0.73785368135098847302126459682, 0,
0.73785368135098847302126459682, 2.58149437427144896698579904662, 3.37202405059832626714624737197, 4.26365735367184896287867240777, 5.44823180660550178627592695111, 6.18409633588450100739691831283, 6.95313247714358247925338231094, 7.85124170632818152646690634321, 8.623076519855784723914258995466
