| L(s) = 1 | + 0.414i·2-s + 2.41i·3-s + 1.82·4-s − 0.999·6-s + 1.58i·8-s − 2.82·9-s + 4.82·11-s + 4.41i·12-s + 0.828i·13-s + 3·16-s + 0.828i·17-s − 1.17i·18-s + 2.82·19-s + 1.99i·22-s − 2.41i·23-s − 3.82·24-s + ⋯ |
| L(s) = 1 | + 0.292i·2-s + 1.39i·3-s + 0.914·4-s − 0.408·6-s + 0.560i·8-s − 0.942·9-s + 1.45·11-s + 1.27i·12-s + 0.229i·13-s + 0.750·16-s + 0.200i·17-s − 0.276i·18-s + 0.648·19-s + 0.426i·22-s − 0.503i·23-s − 0.781·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.275635398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.275635398\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.414iT - 2T^{2} \) |
| 3 | \( 1 - 2.41iT - 3T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 - 0.828iT - 13T^{2} \) |
| 17 | \( 1 - 0.828iT - 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 2.41iT - 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 - 6.41iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + 6.82iT - 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 12.4iT - 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 4.82iT - 73T^{2} \) |
| 79 | \( 1 + 9.17T + 79T^{2} \) |
| 83 | \( 1 + 11.7iT - 83T^{2} \) |
| 89 | \( 1 + 2.65T + 89T^{2} \) |
| 97 | \( 1 + 0.343iT - 97T^{2} \) |
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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
−9.988770069429922394695697018457, −9.246044148871245055439957915539, −8.556118562216916169485907603966, −7.42048995392037357820420422643, −6.60822759061680854314077005328, −5.80475961628193892260178020679, −4.83310560001719782544942422891, −3.90097294873754713993042579872, −3.11088109680775667178349516754, −1.64088987446495908088239160298,
1.07261552949158702836447388854, 1.81010400264682172359812096412, 2.93053675797022498104671371131, 3.96513313544489469295523803248, 5.59415290637284003872352258797, 6.31445448269688669742451737492, 7.15106398583871321836608832740, 7.41811413358286026283107516856, 8.540167598622760281208240986313, 9.439533310999643663955267421104
