L(s) = 1 | + 2-s + 3-s + 4-s + 3.57·5-s + 6-s + 8-s + 9-s + 3.57·10-s + 1.26·11-s + 12-s − 1.93·13-s + 3.57·15-s + 16-s + 6.82·17-s + 18-s − 3.20·19-s + 3.57·20-s + 1.26·22-s + 23-s + 24-s + 7.77·25-s − 1.93·26-s + 27-s − 5.72·29-s + 3.57·30-s − 1.73·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.59·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 1.13·10-s + 0.381·11-s + 0.288·12-s − 0.536·13-s + 0.922·15-s + 0.250·16-s + 1.65·17-s + 0.235·18-s − 0.735·19-s + 0.799·20-s + 0.269·22-s + 0.208·23-s + 0.204·24-s + 1.55·25-s − 0.379·26-s + 0.192·27-s − 1.06·29-s + 0.652·30-s − 0.311·31-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
\(L(1)\) |
\(\approx\) |
\(6.064732538\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.064732538\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.57T + 5T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 13 | \( 1 + 1.93T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 + 3.20T + 19T^{2} \) |
| 29 | \( 1 + 5.72T + 29T^{2} \) |
| 31 | \( 1 + 1.73T + 31T^{2} \) |
| 37 | \( 1 - 9.83T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 3.86T + 43T^{2} \) |
| 47 | \( 1 + 9.68T + 47T^{2} \) |
| 53 | \( 1 - 5.58T + 53T^{2} \) |
| 59 | \( 1 - 8.71T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 + 1.05T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 7.47T + 79T^{2} \) |
| 83 | \( 1 - 1.32T + 83T^{2} \) |
| 89 | \( 1 - 0.524T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−7.74542696745664561187111247899, −7.31986226792035455575610087465, −6.23800858819822282378076824309, −5.96505407440367005090332113552, −5.17959578286337141546237423209, −4.43713055258886416595798706600, −3.48098699440162337563100017445, −2.72374452215430684917796461632, −2.00504866724275198603497248271, −1.22291750505828023569143218399,
1.22291750505828023569143218399, 2.00504866724275198603497248271, 2.72374452215430684917796461632, 3.48098699440162337563100017445, 4.43713055258886416595798706600, 5.17959578286337141546237423209, 5.96505407440367005090332113552, 6.23800858819822282378076824309, 7.31986226792035455575610087465, 7.74542696745664561187111247899