| L(s) = 1 | − 2·2-s − 4-s + 8·8-s − 6·11-s − 7·16-s − 4·17-s + 12·22-s + 12·29-s − 14·32-s + 8·34-s − 12·37-s + 12·41-s + 6·44-s + 12·49-s − 24·58-s + 35·64-s + 4·68-s + 24·74-s − 24·82-s − 28·83-s − 48·88-s − 24·97-s − 24·98-s + 12·101-s + 24·103-s + 28·107-s − 12·116-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s + 2.82·8-s − 1.80·11-s − 7/4·16-s − 0.970·17-s + 2.55·22-s + 2.22·29-s − 2.47·32-s + 1.37·34-s − 1.97·37-s + 1.87·41-s + 0.904·44-s + 12/7·49-s − 3.15·58-s + 35/8·64-s + 0.485·68-s + 2.78·74-s − 2.65·82-s − 3.07·83-s − 5.11·88-s − 2.43·97-s − 2.42·98-s + 1.19·101-s + 2.36·103-s + 2.70·107-s − 1.11·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6133281048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6133281048\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 84 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 88 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−8.927154283222341967053851607442, −8.800528424783218451682875306589, −8.352393382357332741792264618452, −8.220758586343050736642601168415, −7.79135256135885931649247061785, −7.36837153297852668132721529312, −6.91857309965695332087501790355, −6.85052069282560124881582158988, −5.73737794670009027795677272273, −5.71196899919058548607651139890, −5.21857434583651314508262614885, −4.58068510262478333555942227671, −4.41487665622476343538437881058, −4.21191288918414582883401413728, −3.17333382253851467254702560710, −2.98403347034950290298488643006, −2.09793247597275692690779784821, −1.82945342647093792216410892303, −0.72614038227374706708158280756, −0.54248389681717933671431743140,
0.54248389681717933671431743140, 0.72614038227374706708158280756, 1.82945342647093792216410892303, 2.09793247597275692690779784821, 2.98403347034950290298488643006, 3.17333382253851467254702560710, 4.21191288918414582883401413728, 4.41487665622476343538437881058, 4.58068510262478333555942227671, 5.21857434583651314508262614885, 5.71196899919058548607651139890, 5.73737794670009027795677272273, 6.85052069282560124881582158988, 6.91857309965695332087501790355, 7.36837153297852668132721529312, 7.79135256135885931649247061785, 8.220758586343050736642601168415, 8.352393382357332741792264618452, 8.800528424783218451682875306589, 8.927154283222341967053851607442
