| L(s) = 1 | + 8·7-s + 9-s + 12·17-s + 25-s − 16·31-s − 12·41-s + 34·49-s + 8·63-s + 4·73-s − 16·79-s + 81-s + 36·89-s + 4·97-s + 8·103-s − 36·113-s + 96·119-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 12·153-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 3.02·7-s + 1/3·9-s + 2.91·17-s + 1/5·25-s − 2.87·31-s − 1.87·41-s + 34/7·49-s + 1.00·63-s + 0.468·73-s − 1.80·79-s + 1/9·81-s + 3.81·89-s + 0.406·97-s + 0.788·103-s − 3.38·113-s + 8.80·119-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.970·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.846540973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.846540973\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.082506062594783472831133921500, −8.307525687782695431948900879973, −7.965215938754459889106897012614, −7.71052407307967077644420749342, −7.38482725826507043429631860666, −6.76895238282715006434981339956, −5.76555679083117871916592673419, −5.48025531888955534300415720953, −4.95303891523898796353167575120, −4.84143328212506167706839183620, −3.69936628544405197022333195268, −3.62987230449253956202768748221, −2.39041205999551187286748903960, −1.44399343799308961081305068602, −1.43294542786364492056174526795,
1.43294542786364492056174526795, 1.44399343799308961081305068602, 2.39041205999551187286748903960, 3.62987230449253956202768748221, 3.69936628544405197022333195268, 4.84143328212506167706839183620, 4.95303891523898796353167575120, 5.48025531888955534300415720953, 5.76555679083117871916592673419, 6.76895238282715006434981339956, 7.38482725826507043429631860666, 7.71052407307967077644420749342, 7.965215938754459889106897012614, 8.307525687782695431948900879973, 9.082506062594783472831133921500
