L(s) = 1 | + 4·2-s + 16·4-s − 17.4·7-s + 64·8-s + 645.·11-s − 1.09e3·13-s − 69.7·14-s + 256·16-s − 1.16e3·17-s − 2.24e3·19-s + 2.58e3·22-s − 500·23-s − 4.39e3·26-s − 278.·28-s − 470.·29-s + 3.85e3·31-s + 1.02e3·32-s − 4.66e3·34-s + 6.99e3·37-s − 8.97e3·38-s + 9.58e3·41-s − 5.30e3·43-s + 1.03e4·44-s − 2.00e3·46-s − 1.99e4·47-s − 1.65e4·49-s − 1.75e4·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.134·7-s + 0.353·8-s + 1.60·11-s − 1.80·13-s − 0.0950·14-s + 0.250·16-s − 0.978·17-s − 1.42·19-s + 1.13·22-s − 0.197·23-s − 1.27·26-s − 0.0672·28-s − 0.103·29-s + 0.720·31-s + 0.176·32-s − 0.691·34-s + 0.839·37-s − 1.00·38-s + 0.890·41-s − 0.437·43-s + 0.803·44-s − 0.139·46-s − 1.31·47-s − 0.981·49-s − 0.901·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 4T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 17.4T + 1.68e4T^{2} \) |
| 11 | \( 1 - 645.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.09e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.16e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.24e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 500T + 6.43e6T^{2} \) |
| 29 | \( 1 + 470.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.85e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.99e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.58e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.30e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.99e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.14e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.73e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.81e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.62e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.93e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.06e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.69e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.96e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.82e3T + 8.58e9T^{2} \) |
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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
−9.818614998055517722764349907346, −9.061930879948732781601743855471, −7.85234456004308242208786877314, −6.73737122095056696438801894178, −6.23568701966129655737109488399, −4.74132969461172233351228165437, −4.18241215342490192383373159227, −2.79060703526177664011971312297, −1.71923132028723569310754104046, 0,
1.71923132028723569310754104046, 2.79060703526177664011971312297, 4.18241215342490192383373159227, 4.74132969461172233351228165437, 6.23568701966129655737109488399, 6.73737122095056696438801894178, 7.85234456004308242208786877314, 9.061930879948732781601743855471, 9.818614998055517722764349907346