| L(s) = 1 | − 2-s − 3-s + 4-s + 3.26·5-s + 6-s − 8-s + 9-s − 3.26·10-s − 3.52·11-s − 12-s + 1.86·13-s − 3.26·15-s + 16-s + 8.13·17-s − 18-s − 6.86·19-s + 3.26·20-s + 3.52·22-s − 23-s + 24-s + 5.65·25-s − 1.86·26-s − 27-s − 9.17·29-s + 3.26·30-s − 2.47·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.45·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 1.03·10-s − 1.06·11-s − 0.288·12-s + 0.516·13-s − 0.842·15-s + 0.250·16-s + 1.97·17-s − 0.235·18-s − 1.57·19-s + 0.729·20-s + 0.750·22-s − 0.208·23-s + 0.204·24-s + 1.13·25-s − 0.365·26-s − 0.192·27-s − 1.70·29-s + 0.596·30-s − 0.443·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 3.26T + 5T^{2} \) |
| 11 | \( 1 + 3.52T + 11T^{2} \) |
| 13 | \( 1 - 1.86T + 13T^{2} \) |
| 17 | \( 1 - 8.13T + 17T^{2} \) |
| 19 | \( 1 + 6.86T + 19T^{2} \) |
| 29 | \( 1 + 9.17T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 + 4.84T + 37T^{2} \) |
| 41 | \( 1 + 1.54T + 41T^{2} \) |
| 43 | \( 1 - 3.60T + 43T^{2} \) |
| 47 | \( 1 - 1.88T + 47T^{2} \) |
| 53 | \( 1 - 1.87T + 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 + 5.43T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 8.33T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 + 7.99T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−7.59754767127626458532650455210, −6.98405975380823737372204188930, −5.92773796670751283028383034392, −5.81028821922934068455975511315, −5.15070225217024001415424133799, −3.94681939850273370379767207093, −2.91266337973876013428157760535, −1.99897298032722401821388472561, −1.36011913625293452996064715428, 0,
1.36011913625293452996064715428, 1.99897298032722401821388472561, 2.91266337973876013428157760535, 3.94681939850273370379767207093, 5.15070225217024001415424133799, 5.81028821922934068455975511315, 5.92773796670751283028383034392, 6.98405975380823737372204188930, 7.59754767127626458532650455210
