| L(s) = 1 | − 4·4-s + 5·9-s − 66·11-s + 16·16-s + 140·19-s + 450·29-s − 176·31-s − 20·36-s + 864·41-s + 264·44-s − 49·49-s − 960·59-s + 1.62e3·61-s − 64·64-s + 864·71-s − 560·76-s − 850·79-s − 704·81-s − 1.92e3·89-s − 330·99-s − 876·101-s + 3.83e3·109-s − 1.80e3·116-s + 605·121-s + 704·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 5/27·9-s − 1.80·11-s + 1/4·16-s + 1.69·19-s + 2.88·29-s − 1.01·31-s − 0.0925·36-s + 3.29·41-s + 0.904·44-s − 1/7·49-s − 2.11·59-s + 3.40·61-s − 1/8·64-s + 1.44·71-s − 0.845·76-s − 1.21·79-s − 0.965·81-s − 2.28·89-s − 0.335·99-s − 0.863·101-s + 3.36·109-s − 1.44·116-s + 5/11·121-s + 0.509·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.942514545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.942514545\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 p T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2545 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2495 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 22570 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 225 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 88 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 100150 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 432 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 127330 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 38725 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 203510 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 480 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 812 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 246310 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 432 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 649870 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 425 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 198790 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 960 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1322665 T^{2} + p^{6} T^{4} \) |
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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
−11.40985961231806188638372247154, −10.76966040685308886012835996517, −10.24476963401864994185354817956, −9.995752339047791671341062570221, −9.513617207004300963547223004915, −9.017475295201064056858708572042, −8.339658679850305248356127624637, −8.107674714999091627912678035565, −7.45313028963705608528013027553, −7.25563616634511887751888497145, −6.44844333451257791583636340482, −5.76265520629746189179065838504, −5.39350972483079320254787702001, −4.86675750534219107335004466489, −4.36993776404227476984938724357, −3.61767049061658995390571163022, −2.73033807496401543444987247950, −2.59403547171467179085322643698, −1.22113281003954753488476805137, −0.55752770705824908531302636196,
0.55752770705824908531302636196, 1.22113281003954753488476805137, 2.59403547171467179085322643698, 2.73033807496401543444987247950, 3.61767049061658995390571163022, 4.36993776404227476984938724357, 4.86675750534219107335004466489, 5.39350972483079320254787702001, 5.76265520629746189179065838504, 6.44844333451257791583636340482, 7.25563616634511887751888497145, 7.45313028963705608528013027553, 8.107674714999091627912678035565, 8.339658679850305248356127624637, 9.017475295201064056858708572042, 9.513617207004300963547223004915, 9.995752339047791671341062570221, 10.24476963401864994185354817956, 10.76966040685308886012835996517, 11.40985961231806188638372247154
