| L(s) = 1 | + 4·2-s + 16·4-s − 7-s + 64·8-s + 210·11-s − 667·13-s − 4·14-s + 256·16-s − 114·17-s + 581·19-s + 840·22-s + 4.35e3·23-s − 2.66e3·26-s − 16·28-s + 126·29-s + 7.58e3·31-s + 1.02e3·32-s − 456·34-s − 3.74e3·37-s + 2.32e3·38-s + 2.85e3·41-s − 1.82e4·43-s + 3.36e3·44-s + 1.74e4·46-s + 2.33e4·47-s − 1.68e4·49-s − 1.06e4·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.00771·7-s + 0.353·8-s + 0.523·11-s − 1.09·13-s − 0.00545·14-s + 1/4·16-s − 0.0956·17-s + 0.369·19-s + 0.370·22-s + 1.71·23-s − 0.774·26-s − 0.00385·28-s + 0.0278·29-s + 1.41·31-s + 0.176·32-s − 0.0676·34-s − 0.449·37-s + 0.261·38-s + 0.265·41-s − 1.50·43-s + 0.261·44-s + 1.21·46-s + 1.54·47-s − 0.999·49-s − 0.547·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.565096534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.565096534\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + T + p^{5} T^{2} \) |
| 11 | \( 1 - 210 T + p^{5} T^{2} \) |
| 13 | \( 1 + 667 T + p^{5} T^{2} \) |
| 17 | \( 1 + 114 T + p^{5} T^{2} \) |
| 19 | \( 1 - 581 T + p^{5} T^{2} \) |
| 23 | \( 1 - 4350 T + p^{5} T^{2} \) |
| 29 | \( 1 - 126 T + p^{5} T^{2} \) |
| 31 | \( 1 - 7583 T + p^{5} T^{2} \) |
| 37 | \( 1 + 3742 T + p^{5} T^{2} \) |
| 41 | \( 1 - 2856 T + p^{5} T^{2} \) |
| 43 | \( 1 + 18241 T + p^{5} T^{2} \) |
| 47 | \( 1 - 23370 T + p^{5} T^{2} \) |
| 53 | \( 1 - 21684 T + p^{5} T^{2} \) |
| 59 | \( 1 - 32310 T + p^{5} T^{2} \) |
| 61 | \( 1 + 7165 T + p^{5} T^{2} \) |
| 67 | \( 1 - 59579 T + p^{5} T^{2} \) |
| 71 | \( 1 - 43080 T + p^{5} T^{2} \) |
| 73 | \( 1 + 28942 T + p^{5} T^{2} \) |
| 79 | \( 1 - 27608 T + p^{5} T^{2} \) |
| 83 | \( 1 - 1782 T + p^{5} T^{2} \) |
| 89 | \( 1 + 50208 T + p^{5} T^{2} \) |
| 97 | \( 1 - 142793 T + p^{5} T^{2} \) |
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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
−10.34958834644187691339199265327, −9.490141551387591895163782827706, −8.430674848368337284523884346487, −7.24953212998002529779060769731, −6.61136057537245065295319012356, −5.34889189118985118515979248147, −4.60366073576575009616895582279, −3.38223356098737104208396462163, −2.33986754249241014500815885866, −0.894522949625164887922699918163,
0.894522949625164887922699918163, 2.33986754249241014500815885866, 3.38223356098737104208396462163, 4.60366073576575009616895582279, 5.34889189118985118515979248147, 6.61136057537245065295319012356, 7.24953212998002529779060769731, 8.430674848368337284523884346487, 9.490141551387591895163782827706, 10.34958834644187691339199265327
