| L(s) = 1 | − 2-s + 3-s − 2·5-s − 6-s − 2·7-s + 8-s + 2·10-s − 11-s − 2·13-s + 2·14-s − 2·15-s − 16-s − 2·17-s − 6·19-s − 2·21-s + 22-s + 3·23-s + 24-s + 3·25-s + 2·26-s − 27-s + 29-s + 2·30-s − 6·31-s − 33-s + 2·34-s + 4·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.894·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 0.632·10-s − 0.301·11-s − 0.554·13-s + 0.534·14-s − 0.516·15-s − 1/4·16-s − 0.485·17-s − 1.37·19-s − 0.436·21-s + 0.213·22-s + 0.625·23-s + 0.204·24-s + 3/5·25-s + 0.392·26-s − 0.192·27-s + 0.185·29-s + 0.365·30-s − 1.07·31-s − 0.174·33-s + 0.342·34-s + 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6022132833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6022132833\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 5 T - 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 4 T - 55 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−11.49797621387494279074194656732, −10.90937670186892991577168890429, −10.57789869681260290506921332746, −10.12517065453462735269260901831, −9.730605789479653896164563439402, −9.034459442604326415178631316709, −8.862285601515102236885771913953, −8.221564931204832666314638305541, −8.207630922051620251699860032955, −7.24143500776294464532185579915, −6.97485854802334134421766797877, −6.75291627981381086354924312477, −5.65399534387303613384059772834, −5.37145907856310411810976535529, −4.31115519860193369590440338129, −4.15062477912618303856347626263, −3.38417984286778871633795605843, −2.65892728883515571680451632700, −2.05450076650254382549536794668, −0.54642108553476561140389230166,
0.54642108553476561140389230166, 2.05450076650254382549536794668, 2.65892728883515571680451632700, 3.38417984286778871633795605843, 4.15062477912618303856347626263, 4.31115519860193369590440338129, 5.37145907856310411810976535529, 5.65399534387303613384059772834, 6.75291627981381086354924312477, 6.97485854802334134421766797877, 7.24143500776294464532185579915, 8.207630922051620251699860032955, 8.221564931204832666314638305541, 8.862285601515102236885771913953, 9.034459442604326415178631316709, 9.730605789479653896164563439402, 10.12517065453462735269260901831, 10.57789869681260290506921332746, 10.90937670186892991577168890429, 11.49797621387494279074194656732
