| L(s) = 1 | − 2·3-s + 4-s − 5·7-s + 9-s − 2·12-s − 5·13-s + 16-s − 10·19-s + 10·21-s + 7·25-s + 4·27-s − 5·28-s − 12·31-s + 36-s + 13·37-s + 10·39-s − 43-s − 2·48-s + 5·49-s − 5·52-s + 20·57-s − 5·61-s − 5·63-s + 64-s + 2·67-s − 14·73-s − 14·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s − 1.88·7-s + 1/3·9-s − 0.577·12-s − 1.38·13-s + 1/4·16-s − 2.29·19-s + 2.18·21-s + 7/5·25-s + 0.769·27-s − 0.944·28-s − 2.15·31-s + 1/6·36-s + 2.13·37-s + 1.60·39-s − 0.152·43-s − 0.288·48-s + 5/7·49-s − 0.693·52-s + 2.64·57-s − 0.640·61-s − 0.629·63-s + 1/8·64-s + 0.244·67-s − 1.63·73-s − 1.61·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12564 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12564 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 349 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 28 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−10.96022868951970942992846452162, −10.55778947931281726827052425071, −10.11328876384824433590737835300, −9.404854746513586699173069607646, −9.040060130221576340604071239628, −8.141661304865867366216807518388, −7.24043393999108455728733959367, −6.82396896431529593247267188122, −6.26236593286923546634233377226, −5.93420865063290283268366230786, −5.01565263825810260456265325167, −4.28259165008653898965998560229, −3.19327496389644253948195969734, −2.38518738730059013869125023169, 0,
2.38518738730059013869125023169, 3.19327496389644253948195969734, 4.28259165008653898965998560229, 5.01565263825810260456265325167, 5.93420865063290283268366230786, 6.26236593286923546634233377226, 6.82396896431529593247267188122, 7.24043393999108455728733959367, 8.141661304865867366216807518388, 9.040060130221576340604071239628, 9.404854746513586699173069607646, 10.11328876384824433590737835300, 10.55778947931281726827052425071, 10.96022868951970942992846452162
