L(s) = 1 | − 8·4-s + 31.1·7-s + 64·16-s − 155.·19-s − 125·25-s − 249.·28-s + 155.·31-s − 436.·37-s + 520·43-s + 629·49-s − 182·61-s − 512·64-s + 654.·67-s − 374.·73-s + 1.24e3·76-s − 884·79-s + 1.37e3·97-s + 1.00e3·100-s − 1.82e3·103-s − 2.18e3·109-s + 1.99e3·112-s + ⋯ |
L(s) = 1 | − 4-s + 1.68·7-s + 16-s − 1.88·19-s − 25-s − 1.68·28-s + 0.903·31-s − 1.93·37-s + 1.84·43-s + 1.83·49-s − 0.382·61-s − 64-s + 1.19·67-s − 0.599·73-s + 1.88·76-s − 1.25·79-s + 1.43·97-s + 100-s − 1.74·103-s − 1.91·109-s + 1.68·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 8T^{2} \) |
| 5 | \( 1 + 125T^{2} \) |
| 7 | \( 1 - 31.1T + 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 + 155.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 - 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 436.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 520T + 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 + 182T + 2.26e5T^{2} \) |
| 67 | \( 1 - 654.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + 374.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 884T + 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.37e3T + 9.12e5T^{2} \) |
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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
−8.490328606881175717517447574420, −8.216853897304774644156514369066, −7.28937037436413480601284473953, −6.06808775661276535288782590172, −5.21533422105478407613540508452, −4.47156567398099798176236753724, −3.89040580809538059225491106695, −2.30600523739104581527236488882, −1.31732405888759754477880926371, 0,
1.31732405888759754477880926371, 2.30600523739104581527236488882, 3.89040580809538059225491106695, 4.47156567398099798176236753724, 5.21533422105478407613540508452, 6.06808775661276535288782590172, 7.28937037436413480601284473953, 8.216853897304774644156514369066, 8.490328606881175717517447574420