L(s) = 1 | − 2-s + 3-s + 4-s − 3.80·5-s − 6-s − 8-s + 9-s + 3.80·10-s + 5.28·11-s + 12-s − 13-s − 3.80·15-s + 16-s − 6.15·17-s − 18-s − 5.48·19-s − 3.80·20-s − 5.28·22-s + 6.28·23-s − 24-s + 9.48·25-s + 26-s + 27-s − 3·29-s + 3.80·30-s + 8.48·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.70·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 1.20·10-s + 1.59·11-s + 0.288·12-s − 0.277·13-s − 0.982·15-s + 0.250·16-s − 1.49·17-s − 0.235·18-s − 1.25·19-s − 0.850·20-s − 1.12·22-s + 1.31·23-s − 0.204·24-s + 1.89·25-s + 0.196·26-s + 0.192·27-s − 0.557·29-s + 0.694·30-s + 1.52·31-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + 3.80T + 5T^{2} \) |
| 11 | \( 1 - 5.28T + 11T^{2} \) |
| 17 | \( 1 + 6.15T + 17T^{2} \) |
| 19 | \( 1 + 5.48T + 19T^{2} \) |
| 23 | \( 1 - 6.28T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 1.32T + 37T^{2} \) |
| 41 | \( 1 + 5.32T + 41T^{2} \) |
| 43 | \( 1 - 1.61T + 43T^{2} \) |
| 47 | \( 1 + 7.09T + 47T^{2} \) |
| 53 | \( 1 + 0.354T + 53T^{2} \) |
| 59 | \( 1 - 1.67T + 59T^{2} \) |
| 61 | \( 1 - 9.77T + 61T^{2} \) |
| 67 | \( 1 + 8.44T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 + 1.71T + 73T^{2} \) |
| 79 | \( 1 - 2.87T + 79T^{2} \) |
| 83 | \( 1 - 7.44T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 3.15T + 97T^{2} \) |
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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
−8.340175052796634919567290267778, −7.47744110922719956339977541514, −6.80197880383848513442921974159, −6.43328961776142397936612037041, −4.71409163518020764677954635802, −4.20194928363057129949468204277, −3.45537754004240384130782995195, −2.49106358441513258640834254236, −1.26437702323585482963115423198, 0,
1.26437702323585482963115423198, 2.49106358441513258640834254236, 3.45537754004240384130782995195, 4.20194928363057129949468204277, 4.71409163518020764677954635802, 6.43328961776142397936612037041, 6.80197880383848513442921974159, 7.47744110922719956339977541514, 8.340175052796634919567290267778