| L(s) = 1 | − 2-s + 3-s + 4-s − 1.16·5-s − 6-s − 8-s + 9-s + 1.16·10-s + 1.51·11-s + 12-s − 13-s − 1.16·15-s + 16-s − 5.84·17-s − 18-s + 1.45·19-s − 1.16·20-s − 1.51·22-s + 5.77·23-s − 24-s − 3.65·25-s + 26-s + 27-s + 3.59·29-s + 1.16·30-s − 7.29·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.519·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.366·10-s + 0.458·11-s + 0.288·12-s − 0.277·13-s − 0.299·15-s + 0.250·16-s − 1.41·17-s − 0.235·18-s + 0.334·19-s − 0.259·20-s − 0.323·22-s + 1.20·23-s − 0.204·24-s − 0.730·25-s + 0.196·26-s + 0.192·27-s + 0.668·29-s + 0.211·30-s − 1.31·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.406691101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.406691101\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 1.16T + 5T^{2} \) |
| 11 | \( 1 - 1.51T + 11T^{2} \) |
| 17 | \( 1 + 5.84T + 17T^{2} \) |
| 19 | \( 1 - 1.45T + 19T^{2} \) |
| 23 | \( 1 - 5.77T + 23T^{2} \) |
| 29 | \( 1 - 3.59T + 29T^{2} \) |
| 31 | \( 1 + 7.29T + 31T^{2} \) |
| 37 | \( 1 - 1.95T + 37T^{2} \) |
| 41 | \( 1 - 9.88T + 41T^{2} \) |
| 43 | \( 1 + 1.77T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 9.58T + 53T^{2} \) |
| 59 | \( 1 + 0.0156T + 59T^{2} \) |
| 61 | \( 1 + 9.20T + 61T^{2} \) |
| 67 | \( 1 - 4.07T + 67T^{2} \) |
| 71 | \( 1 + 0.358T + 71T^{2} \) |
| 73 | \( 1 + 3.53T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 7.33T + 83T^{2} \) |
| 89 | \( 1 - 9.20T + 89T^{2} \) |
| 97 | \( 1 + 5.19T + 97T^{2} \) |
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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
−8.638008562853616644431597895434, −7.74624107496790659104998288808, −7.24071511040925301689700234494, −6.58336258828906894247390816830, −5.60246793363760235859661494959, −4.50594911920416089376962592856, −3.81743518714587670008878693296, −2.80729987803282385081097595282, −1.99694735389766050658081299196, −0.73656166695898287815410016634,
0.73656166695898287815410016634, 1.99694735389766050658081299196, 2.80729987803282385081097595282, 3.81743518714587670008878693296, 4.50594911920416089376962592856, 5.60246793363760235859661494959, 6.58336258828906894247390816830, 7.24071511040925301689700234494, 7.74624107496790659104998288808, 8.638008562853616644431597895434
