| L(s) = 1 | − 2.30i·2-s + i·3-s − 3.30·4-s + 2.30·6-s − i·7-s + 3.00i·8-s − 9-s − 3·11-s − 3.30i·12-s − 2.60i·13-s − 2.30·14-s + 0.302·16-s − 4.60i·17-s + 2.30i·18-s − 6.60·19-s + ⋯ |
| L(s) = 1 | − 1.62i·2-s + 0.577i·3-s − 1.65·4-s + 0.940·6-s − 0.377i·7-s + 1.06i·8-s − 0.333·9-s − 0.904·11-s − 0.953i·12-s − 0.722i·13-s − 0.615·14-s + 0.0756·16-s − 1.11i·17-s + 0.542i·18-s − 1.51·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.174452 + 0.738993i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.174452 + 0.738993i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 2 | \( 1 + 2.30iT - 2T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 2.60iT - 13T^{2} \) |
| 17 | \( 1 + 4.60iT - 17T^{2} \) |
| 19 | \( 1 + 6.60T + 19T^{2} \) |
| 23 | \( 1 + 6.21iT - 23T^{2} \) |
| 29 | \( 1 - 7.60T + 29T^{2} \) |
| 31 | \( 1 + 7.21T + 31T^{2} \) |
| 37 | \( 1 - 4.21iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 9.60iT - 43T^{2} \) |
| 47 | \( 1 + 10.6iT - 47T^{2} \) |
| 53 | \( 1 - 3.21iT - 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 1.21T + 61T^{2} \) |
| 67 | \( 1 + 15.6iT - 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 - 0.605iT - 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 3.21iT - 83T^{2} \) |
| 89 | \( 1 + 7.81T + 89T^{2} \) |
| 97 | \( 1 - 0.788iT - 97T^{2} \) |
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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
−10.51177179281847902135325567688, −9.913708282843411466407754743097, −8.888040742583642269268543461393, −8.064535135467246990537949870933, −6.61766432208887194782366198082, −5.09070978846423539776304633964, −4.38237457266002397552733504490, −3.19551621889608181635231193347, −2.34238718965055737591782505614, −0.42554655813471938205059797383,
2.15329988216548269930776103203, 4.03275359468556609396536446866, 5.25926885806113168100867426375, 6.01959799402150508290441423897, 6.81718250981032689859539262601, 7.66528625962246294802361989534, 8.452635796123052805120486555803, 9.056488308412033722650964428098, 10.36175880584717710684670964032, 11.38193344424991294539474309506
