| L(s) = 1 | − 2·2-s + 3-s + 3·4-s − 2·6-s − 5·7-s − 4·8-s + 9-s + 3·12-s + 10·14-s + 5·16-s − 2·18-s + 19-s − 5·21-s − 4·24-s − 25-s + 27-s − 15·28-s − 3·29-s − 6·32-s + 3·36-s − 2·38-s + 9·41-s + 10·42-s + 43-s + 5·48-s + 7·49-s + 2·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.816·6-s − 1.88·7-s − 1.41·8-s + 1/3·9-s + 0.866·12-s + 2.67·14-s + 5/4·16-s − 0.471·18-s + 0.229·19-s − 1.09·21-s − 0.816·24-s − 1/5·25-s + 0.192·27-s − 2.83·28-s − 0.557·29-s − 1.06·32-s + 1/2·36-s − 0.324·38-s + 1.40·41-s + 1.54·42-s + 0.152·43-s + 0.721·48-s + 49-s + 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6592624679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6592624679\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
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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
−8.418833611353872124643228840016, −7.78815056927119955040988031098, −7.48122429535089362779962510655, −7.17700152272162333708878409231, −6.59077008283523636029124142405, −6.18984850364578817872960351691, −5.93242350461445619756406693256, −5.23441031610075988010423822976, −4.42109395514014922756054503338, −3.72771342196013460624801741236, −3.35615304191926574093088690732, −2.76292330274880816941565421966, −2.33362143873072042261103300681, −1.45328346897671337588415246371, −0.48552472125052931553080682251,
0.48552472125052931553080682251, 1.45328346897671337588415246371, 2.33362143873072042261103300681, 2.76292330274880816941565421966, 3.35615304191926574093088690732, 3.72771342196013460624801741236, 4.42109395514014922756054503338, 5.23441031610075988010423822976, 5.93242350461445619756406693256, 6.18984850364578817872960351691, 6.59077008283523636029124142405, 7.17700152272162333708878409231, 7.48122429535089362779962510655, 7.78815056927119955040988031098, 8.418833611353872124643228840016
