| L(s) = 1 | − 5-s + 5·13-s − 17-s − 9·25-s + 5·29-s − 4·37-s − 6·41-s − 3·49-s − 53-s − 9·61-s − 5·65-s + 85-s + 24·89-s + 21·101-s + 3·109-s − 6·121-s + 14·125-s + 127-s + 131-s + 137-s + 139-s − 5·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.38·13-s − 0.242·17-s − 9/5·25-s + 0.928·29-s − 0.657·37-s − 0.937·41-s − 3/7·49-s − 0.137·53-s − 1.15·61-s − 0.620·65-s + 0.108·85-s + 2.54·89-s + 2.08·101-s + 0.287·109-s − 0.545·121-s + 1.25·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.415·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1492992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1492992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 120 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−7.75131015270102957838703805125, −7.41761956182991062155450451576, −6.66570079078758190202339775940, −6.46087015529664351575420700946, −5.99660151021143095834813309697, −5.61521787458339036697118744610, −4.93069139816258707626616899881, −4.59849801425569271579626538356, −3.99293621036674839221366213302, −3.48341474914188442649788065880, −3.30562631646835554505947825879, −2.36131156617102498816789393014, −1.80461179121259826567587182813, −1.08100905498209309148644806452, 0,
1.08100905498209309148644806452, 1.80461179121259826567587182813, 2.36131156617102498816789393014, 3.30562631646835554505947825879, 3.48341474914188442649788065880, 3.99293621036674839221366213302, 4.59849801425569271579626538356, 4.93069139816258707626616899881, 5.61521787458339036697118744610, 5.99660151021143095834813309697, 6.46087015529664351575420700946, 6.66570079078758190202339775940, 7.41761956182991062155450451576, 7.75131015270102957838703805125
