| L(s) = 1 | + 2·2-s − 6·3-s + 6·4-s − 12·6-s − 26·7-s + 32·8-s + 27·9-s + 28·11-s − 36·12-s − 18·13-s − 52·14-s + 44·16-s + 68·17-s + 54·18-s + 6·19-s + 156·21-s + 56·22-s − 132·23-s − 192·24-s − 36·26-s − 108·27-s − 156·28-s + 92·29-s + 122·31-s + 120·32-s − 168·33-s + 136·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 3/4·4-s − 0.816·6-s − 1.40·7-s + 1.41·8-s + 9-s + 0.767·11-s − 0.866·12-s − 0.384·13-s − 0.992·14-s + 0.687·16-s + 0.970·17-s + 0.707·18-s + 0.0724·19-s + 1.62·21-s + 0.542·22-s − 1.19·23-s − 1.63·24-s − 0.271·26-s − 0.769·27-s − 1.05·28-s + 0.589·29-s + 0.706·31-s + 0.662·32-s − 0.886·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.968256539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.968256539\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - p T - p T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 26 T + 551 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 28 T + 2554 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 18 T - 389 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 p T + 3382 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 13423 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 132 T + 25954 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 92 T + 35998 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 122 T + 48407 T^{2} - 122 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 284 T + 110526 T^{2} - 284 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 392 T + 124882 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 690 T + 277735 T^{2} + 690 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 620 T + 284290 T^{2} - 620 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 p T + 462634 T^{2} - 16 p^{4} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 124 T + 336778 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 750 T + 535003 T^{2} - 750 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 358 T + 443567 T^{2} - 358 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 824 T + 877966 T^{2} - 824 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 108 T + 779734 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 880 T + 693278 T^{2} + 880 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 156 T + 449242 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 864 T + 1202578 T^{2} + 864 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 521 T + p^{3} T^{2} )^{2} \) |
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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
−14.03788744922045041459937948761, −13.88639904392825721058657660870, −12.89573094209689466971672645536, −12.84155207522660950040645693125, −12.09777727215669085354721759107, −11.64513536298935963850060994379, −11.28826108147487817190758028332, −10.29743689672137619265614504293, −10.05857301981669474608956321730, −9.658630696943240842892968042720, −8.527440066004152581283656322330, −7.59758035965249877213244615909, −7.01964205401379894543070445435, −6.47057889943727675642661651267, −5.94500376126656922686914867451, −5.23797604469871097058329697238, −4.29056976609640423991413652363, −3.71957028389663821607026622703, −2.42335879264394898791453723953, −0.911195730578247287836466304871,
0.911195730578247287836466304871, 2.42335879264394898791453723953, 3.71957028389663821607026622703, 4.29056976609640423991413652363, 5.23797604469871097058329697238, 5.94500376126656922686914867451, 6.47057889943727675642661651267, 7.01964205401379894543070445435, 7.59758035965249877213244615909, 8.527440066004152581283656322330, 9.658630696943240842892968042720, 10.05857301981669474608956321730, 10.29743689672137619265614504293, 11.28826108147487817190758028332, 11.64513536298935963850060994379, 12.09777727215669085354721759107, 12.84155207522660950040645693125, 12.89573094209689466971672645536, 13.88639904392825721058657660870, 14.03788744922045041459937948761
