| L(s) = 1 | + 6·11-s − 13-s − 4·17-s − 4·19-s + 4·23-s − 4·29-s − 8·31-s + 6·37-s − 10·41-s − 12·43-s − 2·47-s − 7·49-s + 12·53-s + 2·59-s − 14·61-s + 4·67-s + 6·71-s − 10·73-s + 8·79-s + 6·83-s + 2·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.80·11-s − 0.277·13-s − 0.970·17-s − 0.917·19-s + 0.834·23-s − 0.742·29-s − 1.43·31-s + 0.986·37-s − 1.56·41-s − 1.82·43-s − 0.291·47-s − 49-s + 1.64·53-s + 0.260·59-s − 1.79·61-s + 0.488·67-s + 0.712·71-s − 1.17·73-s + 0.900·79-s + 0.658·83-s + 0.211·89-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.733656179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.733656179\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.73782529369757, −14.16589218122795, −13.44154280965481, −13.16305118846812, −12.55457112309818, −11.95755366375685, −11.45271866665516, −11.13028655942412, −10.48463336186047, −9.828478658374078, −9.235390714329209, −8.931114754156276, −8.412604819412846, −7.687172594250814, −6.833021046639143, −6.760855510603637, −6.136860278147274, −5.331005592256887, −4.747332846193745, −4.089288559329077, −3.638006012896032, −2.917527136189992, −1.874055658718333, −1.620967327458397, −0.4568546763258845,
0.4568546763258845, 1.620967327458397, 1.874055658718333, 2.917527136189992, 3.638006012896032, 4.089288559329077, 4.747332846193745, 5.331005592256887, 6.136860278147274, 6.760855510603637, 6.833021046639143, 7.687172594250814, 8.412604819412846, 8.931114754156276, 9.235390714329209, 9.828478658374078, 10.48463336186047, 11.13028655942412, 11.45271866665516, 11.95755366375685, 12.55457112309818, 13.16305118846812, 13.44154280965481, 14.16589218122795, 14.73782529369757
