| L(s) = 1 | − 3-s + 4-s + 3·5-s − 5·7-s − 12-s − 3·15-s + 16-s + 3·20-s + 5·21-s − 25-s + 4·27-s − 5·28-s + 12·29-s + 31-s − 15·35-s + 43-s − 48-s + 7·49-s − 3·60-s + 10·61-s + 64-s + 18·71-s + 28·73-s + 75-s + 3·80-s − 7·81-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s + 1.34·5-s − 1.88·7-s − 0.288·12-s − 0.774·15-s + 1/4·16-s + 0.670·20-s + 1.09·21-s − 1/5·25-s + 0.769·27-s − 0.944·28-s + 2.22·29-s + 0.179·31-s − 2.53·35-s + 0.152·43-s − 0.144·48-s + 49-s − 0.387·60-s + 1.28·61-s + 1/8·64-s + 2.13·71-s + 3.27·73-s + 0.115·75-s + 0.335·80-s − 7/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142572 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142572 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.457925107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.457925107\) |
| \(L(\frac{3}{2})\) |
|
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_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 109 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.549221074433231587243108504446, −9.031380500310387501594431559723, −8.334920033549714665181303782023, −7.948876855967197137814838448268, −6.94466207806107708542261581167, −6.71809962931763807637595342800, −6.30881141731258861146971996298, −6.06604012604939495367968420978, −5.36862204443752991445071036588, −4.96171280110319348373087981166, −3.99593192770836022605704366368, −3.31309158468217631442480663082, −2.68520442386714259414545395486, −2.11746691331465119880445539709, −0.846424080309684912971905626162,
0.846424080309684912971905626162, 2.11746691331465119880445539709, 2.68520442386714259414545395486, 3.31309158468217631442480663082, 3.99593192770836022605704366368, 4.96171280110319348373087981166, 5.36862204443752991445071036588, 6.06604012604939495367968420978, 6.30881141731258861146971996298, 6.71809962931763807637595342800, 6.94466207806107708542261581167, 7.948876855967197137814838448268, 8.334920033549714665181303782023, 9.031380500310387501594431559723, 9.549221074433231587243108504446
