| L(s) = 1 | + 2·5-s + 2·9-s + 2·13-s + 4·17-s + 3·25-s − 4·29-s − 4·37-s + 4·41-s + 4·45-s + 2·49-s + 12·53-s − 4·61-s + 4·65-s + 4·73-s − 5·81-s + 8·85-s + 4·89-s − 12·97-s + 12·101-s + 12·109-s − 12·113-s + 4·117-s + 18·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2/3·9-s + 0.554·13-s + 0.970·17-s + 3/5·25-s − 0.742·29-s − 0.657·37-s + 0.624·41-s + 0.596·45-s + 2/7·49-s + 1.64·53-s − 0.512·61-s + 0.496·65-s + 0.468·73-s − 5/9·81-s + 0.867·85-s + 0.423·89-s − 1.21·97-s + 1.19·101-s + 1.14·109-s − 1.12·113-s + 0.369·117-s + 1.63·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.695035714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.695035714\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.546465749511268910415782865259, −7.987109424737057245307656397071, −7.53125199949556523939117458653, −7.09717057872848487510621024148, −6.68862159372961218064790446970, −6.08902705980528534510708629881, −5.67575443421577017654436013888, −5.37529001107628665578409038673, −4.71703251013746342009376808605, −4.13991613053845340278766409570, −3.60795635925484158023154455185, −3.03540755191807680960141033038, −2.25957846055533507593722255797, −1.65967151690953797211524541958, −0.919009355606882619009403181064,
0.919009355606882619009403181064, 1.65967151690953797211524541958, 2.25957846055533507593722255797, 3.03540755191807680960141033038, 3.60795635925484158023154455185, 4.13991613053845340278766409570, 4.71703251013746342009376808605, 5.37529001107628665578409038673, 5.67575443421577017654436013888, 6.08902705980528534510708629881, 6.68862159372961218064790446970, 7.09717057872848487510621024148, 7.53125199949556523939117458653, 7.987109424737057245307656397071, 8.546465749511268910415782865259
