| L(s) = 1 | + 2·3-s + 3·9-s − 6·13-s + 16·17-s + 12·23-s − 25-s + 4·27-s − 16·29-s − 12·39-s + 8·43-s − 2·49-s + 32·51-s − 20·53-s − 4·61-s + 24·69-s − 2·75-s − 16·79-s + 5·81-s − 32·87-s − 8·101-s − 8·103-s − 32·113-s − 18·117-s + 6·121-s + 127-s + 16·129-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 1.66·13-s + 3.88·17-s + 2.50·23-s − 1/5·25-s + 0.769·27-s − 2.97·29-s − 1.92·39-s + 1.21·43-s − 2/7·49-s + 4.48·51-s − 2.74·53-s − 0.512·61-s + 2.88·69-s − 0.230·75-s − 1.80·79-s + 5/9·81-s − 3.43·87-s − 0.796·101-s − 0.788·103-s − 3.01·113-s − 1.66·117-s + 6/11·121-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.361847621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.361847621\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.066308702369827409553621163387, −8.220666797236457858746231767366, −8.200110282383145890866493155853, −7.73202441239916683449209436850, −7.33107992865207975730835174616, −7.26174295638853062074866197758, −7.01647037065046332282196338199, −6.14862255037702206953111810004, −5.71096803569977445025242150173, −5.28653482184696191354381013674, −5.28138785358295123935793752687, −4.51004254909634017364652854761, −4.22753486122357343453267840212, −3.41807567641843048252724896440, −3.16645393952271746742943028343, −3.12518059792423395302813596509, −2.45935141114399907011436111127, −1.66431058912171513783871927535, −1.43965731527577456337682594488, −0.61258504852254868490751734947,
0.61258504852254868490751734947, 1.43965731527577456337682594488, 1.66431058912171513783871927535, 2.45935141114399907011436111127, 3.12518059792423395302813596509, 3.16645393952271746742943028343, 3.41807567641843048252724896440, 4.22753486122357343453267840212, 4.51004254909634017364652854761, 5.28138785358295123935793752687, 5.28653482184696191354381013674, 5.71096803569977445025242150173, 6.14862255037702206953111810004, 7.01647037065046332282196338199, 7.26174295638853062074866197758, 7.33107992865207975730835174616, 7.73202441239916683449209436850, 8.200110282383145890866493155853, 8.220666797236457858746231767366, 9.066308702369827409553621163387
