L(s) = 1 | − 29.2·3-s + 7.45·5-s + 613.·9-s − 447.·11-s − 383.·13-s − 218.·15-s + 1.87e3·17-s + 7.59·19-s − 2.69e3·23-s − 3.06e3·25-s − 1.08e4·27-s − 8.77e3·29-s + 7.54e3·31-s + 1.31e4·33-s + 6.51e3·37-s + 1.12e4·39-s + 1.79e4·41-s − 1.35e3·43-s + 4.57e3·45-s + 1.82e3·47-s − 5.48e4·51-s + 3.58e4·53-s − 3.33e3·55-s − 222.·57-s − 2.36e4·59-s − 2.70e3·61-s − 2.85e3·65-s + ⋯ |
L(s) = 1 | − 1.87·3-s + 0.133·5-s + 2.52·9-s − 1.11·11-s − 0.628·13-s − 0.250·15-s + 1.57·17-s + 0.00482·19-s − 1.06·23-s − 0.982·25-s − 2.86·27-s − 1.93·29-s + 1.41·31-s + 2.09·33-s + 0.782·37-s + 1.18·39-s + 1.67·41-s − 0.111·43-s + 0.337·45-s + 0.120·47-s − 2.95·51-s + 1.75·53-s − 0.148·55-s − 0.00905·57-s − 0.883·59-s − 0.0929·61-s − 0.0839·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 29.2T + 243T^{2} \) |
| 5 | \( 1 - 7.45T + 3.12e3T^{2} \) |
| 11 | \( 1 + 447.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 383.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.87e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 7.59T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.69e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.77e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.54e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.51e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.79e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.35e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.82e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.58e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.36e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.70e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.68e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.37e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.69e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.44e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.24e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.18e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.67e4T + 8.58e9T^{2} \) |
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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
−9.618694347366780429754166824015, −7.79358340666505403726708502963, −7.45230159646695431408595503068, −6.09080014010807144704541578726, −5.68673931290751840621583355048, −4.90541552089148785401146733833, −3.87844052129769436528324352576, −2.23037751757799508906303591818, −0.917115402185362505151887590598, 0,
0.917115402185362505151887590598, 2.23037751757799508906303591818, 3.87844052129769436528324352576, 4.90541552089148785401146733833, 5.68673931290751840621583355048, 6.09080014010807144704541578726, 7.45230159646695431408595503068, 7.79358340666505403726708502963, 9.618694347366780429754166824015