| L(s) = 1 | + 2·11-s − 4·16-s + 4·19-s − 4·29-s − 12·31-s + 10·41-s + 5·49-s + 16·59-s + 26·61-s + 10·71-s + 6·79-s − 30·89-s + 32·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s − 8·176-s + 179-s + ⋯ |
| L(s) = 1 | + 0.603·11-s − 16-s + 0.917·19-s − 0.742·29-s − 2.15·31-s + 1.56·41-s + 5/7·49-s + 2.08·59-s + 3.32·61-s + 1.18·71-s + 0.675·79-s − 3.17·89-s + 3.06·109-s − 1.72·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.0769·169-s + 0.0760·173-s − 0.603·176-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.399735117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.399735117\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
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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
−9.037008400611670573419571767770, −8.538322671737764831684737668083, −8.269985654050410196391348434937, −7.84435811853760561171427918230, −7.19744131751515319901032682505, −7.07978112278912470948974670538, −6.92908566360159277454193116003, −6.32810379195930940007664517600, −5.70001926955390453610685032465, −5.50068394414761151734445971764, −5.31213003662936222885185532060, −4.55577859397052027099335077742, −4.20855102704911379404082232144, −3.71648278380554459268328947114, −3.55103952631971567866707456847, −2.81377589737374436834965191329, −2.13667853204613269870672906780, −2.06356156990502496368370980074, −1.11616088887264556073464043862, −0.54684343669856144119168045931,
0.54684343669856144119168045931, 1.11616088887264556073464043862, 2.06356156990502496368370980074, 2.13667853204613269870672906780, 2.81377589737374436834965191329, 3.55103952631971567866707456847, 3.71648278380554459268328947114, 4.20855102704911379404082232144, 4.55577859397052027099335077742, 5.31213003662936222885185532060, 5.50068394414761151734445971764, 5.70001926955390453610685032465, 6.32810379195930940007664517600, 6.92908566360159277454193116003, 7.07978112278912470948974670538, 7.19744131751515319901032682505, 7.84435811853760561171427918230, 8.269985654050410196391348434937, 8.538322671737764831684737668083, 9.037008400611670573419571767770
