| L(s) = 1 | − 4·4-s − 24·11-s + 16·16-s + 200·19-s − 180·29-s + 304·31-s + 876·41-s + 96·44-s + 670·49-s + 840·59-s + 1.80e3·61-s − 64·64-s − 864·71-s − 800·76-s + 320·79-s + 1.62e3·89-s + 516·101-s − 1.90e3·109-s + 720·116-s − 2.23e3·121-s − 1.21e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.657·11-s + 1/4·16-s + 2.41·19-s − 1.15·29-s + 1.76·31-s + 3.33·41-s + 0.328·44-s + 1.95·49-s + 1.85·59-s + 3.78·61-s − 1/8·64-s − 1.44·71-s − 1.20·76-s + 0.455·79-s + 1.92·89-s + 0.508·101-s − 1.66·109-s + 0.576·116-s − 1.67·121-s − 0.880·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.953327014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.953327014\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 670 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 12 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1030 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5470 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 100 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6910 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 90 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 152 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 100150 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 438 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 157990 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 166030 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 248470 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 420 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 902 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 447050 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 432 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 646990 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 160 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1138390 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 810 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 602110 T^{2} + p^{6} T^{4} \) |
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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
−10.72848523748022034471888247187, −10.51804494624323660398869690701, −9.867074630777899289913721191661, −9.560657420168125583345953404815, −9.241127417908519745188081306337, −8.671545843867980458318146727698, −8.019383443235623041770190810278, −7.85307847814056745705920043101, −7.09416946760370309077869307043, −7.05372382141228285960933022142, −5.93781691166390590797326251258, −5.68093892312792666205441509627, −5.26132311540951957578189144581, −4.65756054180351006520017865160, −3.98117166468712184266240546502, −3.56924481079303792323852415316, −2.67582355966347299025569646735, −2.34921737102649215019836779592, −0.959846312424796667948307735738, −0.75602300209059065478118762817,
0.75602300209059065478118762817, 0.959846312424796667948307735738, 2.34921737102649215019836779592, 2.67582355966347299025569646735, 3.56924481079303792323852415316, 3.98117166468712184266240546502, 4.65756054180351006520017865160, 5.26132311540951957578189144581, 5.68093892312792666205441509627, 5.93781691166390590797326251258, 7.05372382141228285960933022142, 7.09416946760370309077869307043, 7.85307847814056745705920043101, 8.019383443235623041770190810278, 8.671545843867980458318146727698, 9.241127417908519745188081306337, 9.560657420168125583345953404815, 9.867074630777899289913721191661, 10.51804494624323660398869690701, 10.72848523748022034471888247187
