| L(s) = 1 | − i·3-s + 1.41i·5-s − 1.41·7-s − 9-s + 2i·11-s + 1.41·15-s + 2·17-s − 4i·19-s + 1.41i·21-s − 2.82·23-s + 2.99·25-s + i·27-s + 9.89i·29-s − 7.07·31-s + 2·33-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.632i·5-s − 0.534·7-s − 0.333·9-s + 0.603i·11-s + 0.365·15-s + 0.485·17-s − 0.917i·19-s + 0.308i·21-s − 0.589·23-s + 0.599·25-s + 0.192i·27-s + 1.83i·29-s − 1.27·31-s + 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.021514864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.021514864\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 9.89iT - 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 - 8.48iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 1.41iT - 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 - 14.1iT - 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−9.635373431652891511408726279400, −8.878023410003211907563164867934, −7.943255579253890407371257929761, −6.97630678620849124897716706218, −6.76241129180993567421761964775, −5.65554863927305572843090020217, −4.72164577123277532950718553295, −3.41813048048615366335659041423, −2.69829836843515740546085507043, −1.41319520582790997952087696415,
0.40478530864035719151949998930, 2.05628216994057905664106507850, 3.44815816137743851729648616350, 4.01186745391018928641963689062, 5.22487560292815613022424297973, 5.78573880930752368357860752782, 6.72864005944816266343006699796, 7.914478324494061593336695453263, 8.455091562835493959832636893074, 9.422792515214011195298800979943
