| L(s) = 1 | − 2·7-s + 6·9-s − 12·17-s + 6·25-s − 16·31-s − 4·41-s + 16·47-s + 3·49-s − 12·63-s − 16·71-s − 20·73-s − 32·79-s + 27·81-s + 12·89-s − 12·97-s − 32·103-s + 4·113-s + 24·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 72·153-s + 157-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 2·9-s − 2.91·17-s + 6/5·25-s − 2.87·31-s − 0.624·41-s + 2.33·47-s + 3/7·49-s − 1.51·63-s − 1.89·71-s − 2.34·73-s − 3.60·79-s + 3·81-s + 1.27·89-s − 1.21·97-s − 3.15·103-s + 0.376·113-s + 2.20·119-s + 6/11·121-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 − 5.82·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.318146764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.318146764\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−9.745836656439833316282704051939, −8.903712408776508411217315203105, −8.890938324018158262513287189505, −8.582266227698537095806042349993, −7.70571146092749935795184798176, −7.25214299997128106071549019140, −7.11196304633871858201230720088, −6.84466491763944435938810317878, −6.50588879532471922078838354857, −5.68784206126345907746742174977, −5.67284219814268850127895572559, −4.78320428605617232983467798508, −4.42345508652110506433275110732, −4.13262069038847401481089269234, −3.83905909196373736255552550968, −2.98388876822502447763400489625, −2.61355728361102460213543875210, −1.69115921376259029597674795708, −1.67703211091000580964232125707, −0.41698006543836759305244622919,
0.41698006543836759305244622919, 1.67703211091000580964232125707, 1.69115921376259029597674795708, 2.61355728361102460213543875210, 2.98388876822502447763400489625, 3.83905909196373736255552550968, 4.13262069038847401481089269234, 4.42345508652110506433275110732, 4.78320428605617232983467798508, 5.67284219814268850127895572559, 5.68784206126345907746742174977, 6.50588879532471922078838354857, 6.84466491763944435938810317878, 7.11196304633871858201230720088, 7.25214299997128106071549019140, 7.70571146092749935795184798176, 8.582266227698537095806042349993, 8.890938324018158262513287189505, 8.903712408776508411217315203105, 9.745836656439833316282704051939
