| L(s) = 1 | − 4-s + 2·8-s + 2·9-s − 8·11-s + 16-s − 9·23-s + 2·25-s + 4·29-s − 4·32-s − 2·36-s + 4·37-s + 8·44-s − 7·49-s + 12·53-s + 3·64-s + 8·67-s + 8·71-s + 4·72-s + 16·79-s − 5·81-s − 16·88-s + 9·92-s − 16·99-s − 2·100-s + 16·107-s − 4·109-s − 20·113-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.707·8-s + 2/3·9-s − 2.41·11-s + 1/4·16-s − 1.87·23-s + 2/5·25-s + 0.742·29-s − 0.707·32-s − 1/3·36-s + 0.657·37-s + 1.20·44-s − 49-s + 1.64·53-s + 3/8·64-s + 0.977·67-s + 0.949·71-s + 0.471·72-s + 1.80·79-s − 5/9·81-s − 1.70·88-s + 0.938·92-s − 1.60·99-s − 1/5·100-s + 1.54·107-s − 0.383·109-s − 1.88·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6141994056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6141994056\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 8 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 142 T^{2} + p^{2} T^{4} \) |
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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
−13.09934366291682023430039040733, −12.79059384436159602278559496449, −12.13240782852281376960972270117, −11.21358949587687420939859217756, −10.46423130684701605502983097995, −10.20840258377203912940977449231, −9.638040636338315315306407369116, −8.536015900210194161224770261535, −7.901065581287480949315837392901, −7.59842438501720390720393292269, −6.51957365678974001666102365983, −5.43404419368717190931601781158, −4.86525372116713891728704442672, −3.91672060964165073859091986551, −2.41431212470637847961403026107,
2.41431212470637847961403026107, 3.91672060964165073859091986551, 4.86525372116713891728704442672, 5.43404419368717190931601781158, 6.51957365678974001666102365983, 7.59842438501720390720393292269, 7.901065581287480949315837392901, 8.536015900210194161224770261535, 9.638040636338315315306407369116, 10.20840258377203912940977449231, 10.46423130684701605502983097995, 11.21358949587687420939859217756, 12.13240782852281376960972270117, 12.79059384436159602278559496449, 13.09934366291682023430039040733
