L(s) = 1 | − 2-s − 3-s + 4-s − 2.89·5-s + 6-s − 8-s + 9-s + 2.89·10-s + 1.61·11-s − 12-s − 3.02·13-s + 2.89·15-s + 16-s + 5.54·17-s − 18-s + 2.76·19-s − 2.89·20-s − 1.61·22-s + 23-s + 24-s + 3.39·25-s + 3.02·26-s − 27-s + 1.76·29-s − 2.89·30-s − 1.60·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.29·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.916·10-s + 0.488·11-s − 0.288·12-s − 0.839·13-s + 0.748·15-s + 0.250·16-s + 1.34·17-s − 0.235·18-s + 0.635·19-s − 0.647·20-s − 0.345·22-s + 0.208·23-s + 0.204·24-s + 0.678·25-s + 0.593·26-s − 0.192·27-s + 0.328·29-s − 0.528·30-s − 0.288·31-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
\(L(1)\) |
\(\approx\) |
\(0.7154495005\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7154495005\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.89T + 5T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 + 3.02T + 13T^{2} \) |
| 17 | \( 1 - 5.54T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 29 | \( 1 - 1.76T + 29T^{2} \) |
| 31 | \( 1 + 1.60T + 31T^{2} \) |
| 37 | \( 1 + 1.02T + 37T^{2} \) |
| 41 | \( 1 - 6.76T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 4.15T + 47T^{2} \) |
| 53 | \( 1 - 4.89T + 53T^{2} \) |
| 59 | \( 1 + 6.95T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 7.23T + 67T^{2} \) |
| 71 | \( 1 - 4.87T + 71T^{2} \) |
| 73 | \( 1 - 5.13T + 73T^{2} \) |
| 79 | \( 1 + 6.89T + 79T^{2} \) |
| 83 | \( 1 - 5.15T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−7.79917949123082815069473486362, −7.45565235505272539113962156357, −6.82968459598738988152855361980, −5.92835405199062548723762131934, −5.18297054275459945096255831533, −4.35315445871337384944617272902, −3.56589061479397468092632003167, −2.78948479977735512286874614245, −1.43748968176287777293829702175, −0.53366150296210084484886168875,
0.53366150296210084484886168875, 1.43748968176287777293829702175, 2.78948479977735512286874614245, 3.56589061479397468092632003167, 4.35315445871337384944617272902, 5.18297054275459945096255831533, 5.92835405199062548723762131934, 6.82968459598738988152855361980, 7.45565235505272539113962156357, 7.79917949123082815069473486362