| L(s) = 1 | + 2·7-s − 2·9-s + 12·23-s + 25-s + 12·29-s + 4·37-s − 20·43-s − 3·49-s − 12·53-s − 4·63-s + 4·67-s − 24·71-s + 16·79-s − 5·81-s − 12·107-s + 4·109-s − 12·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 24·161-s + 163-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 2/3·9-s + 2.50·23-s + 1/5·25-s + 2.22·29-s + 0.657·37-s − 3.04·43-s − 3/7·49-s − 1.64·53-s − 0.503·63-s + 0.488·67-s − 2.84·71-s + 1.80·79-s − 5/9·81-s − 1.16·107-s + 0.383·109-s − 1.12·113-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 + 0.0798·157-s + 1.89·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.213850982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.213850982\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−10.83473950301065015207539180976, −10.54734551724404681829893357471, −9.789054880187459544551674749578, −9.237629786112200017815782266993, −8.552217550781204179646223792184, −8.321958701563865500343115449553, −7.67946935430715687873464278567, −6.77106114591073562021381363331, −6.57891116465648258947670054106, −5.60226623689013744024577168227, −4.78130792717525308450176413839, −4.73531593841690414562612515004, −3.28633294922749820286908619224, −2.80176924686893785247315174730, −1.37767620904697515455020569610,
1.37767620904697515455020569610, 2.80176924686893785247315174730, 3.28633294922749820286908619224, 4.73531593841690414562612515004, 4.78130792717525308450176413839, 5.60226623689013744024577168227, 6.57891116465648258947670054106, 6.77106114591073562021381363331, 7.67946935430715687873464278567, 8.321958701563865500343115449553, 8.552217550781204179646223792184, 9.237629786112200017815782266993, 9.789054880187459544551674749578, 10.54734551724404681829893357471, 10.83473950301065015207539180976
