L(s) = 1 | + 3-s − 4·7-s + 9-s + 12·13-s − 4·21-s − 10·25-s + 27-s − 20·31-s + 4·37-s + 12·39-s − 16·43-s − 2·49-s + 4·61-s − 4·63-s − 8·67-s − 20·73-s − 10·75-s + 12·79-s + 81-s − 48·91-s − 20·93-s − 12·97-s + 20·103-s − 4·109-s + 4·111-s + 12·117-s − 6·121-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s + 3.32·13-s − 0.872·21-s − 2·25-s + 0.192·27-s − 3.59·31-s + 0.657·37-s + 1.92·39-s − 2.43·43-s − 2/7·49-s + 0.512·61-s − 0.503·63-s − 0.977·67-s − 2.34·73-s − 1.15·75-s + 1.35·79-s + 1/9·81-s − 5.03·91-s − 2.07·93-s − 1.21·97-s + 1.97·103-s − 0.383·109-s + 0.379·111-s + 1.10·117-s − 0.545·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.658903060264415544055202217864, −7.76841700352222764456930738236, −7.70036027517747780731408099423, −6.81542041238937589948552667187, −6.52243750033071865852983380426, −6.04519620219891673101591163967, −5.75568326693151721019291253980, −5.19382499916710112913058943849, −3.97633443717640489814889299763, −3.94434944725762226728078180666, −3.36330701716196318281397292070, −3.13420541549434036877759112013, −1.86962407001750988737926530376, −1.50413098873776121963751941726, 0,
1.50413098873776121963751941726, 1.86962407001750988737926530376, 3.13420541549434036877759112013, 3.36330701716196318281397292070, 3.94434944725762226728078180666, 3.97633443717640489814889299763, 5.19382499916710112913058943849, 5.75568326693151721019291253980, 6.04519620219891673101591163967, 6.52243750033071865852983380426, 6.81542041238937589948552667187, 7.70036027517747780731408099423, 7.76841700352222764456930738236, 8.658903060264415544055202217864