L(s) = 1 | − 2.24·2-s + 3-s + 3.02·4-s − 1.20·5-s − 2.24·6-s − 7-s − 2.30·8-s + 9-s + 2.69·10-s + 3.48·11-s + 3.02·12-s + 3.78·13-s + 2.24·14-s − 1.20·15-s − 0.894·16-s + 3.70·17-s − 2.24·18-s + 6.98·19-s − 3.64·20-s − 21-s − 7.80·22-s + 3.91·23-s − 2.30·24-s − 3.55·25-s − 8.48·26-s + 27-s − 3.02·28-s + ⋯ |
L(s) = 1 | − 1.58·2-s + 0.577·3-s + 1.51·4-s − 0.538·5-s − 0.915·6-s − 0.377·7-s − 0.813·8-s + 0.333·9-s + 0.853·10-s + 1.05·11-s + 0.873·12-s + 1.04·13-s + 0.599·14-s − 0.310·15-s − 0.223·16-s + 0.898·17-s − 0.528·18-s + 1.60·19-s − 0.814·20-s − 0.218·21-s − 1.66·22-s + 0.816·23-s − 0.469·24-s − 0.710·25-s − 1.66·26-s + 0.192·27-s − 0.571·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.214135595\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.214135595\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 5 | \( 1 + 1.20T + 5T^{2} \) |
| 11 | \( 1 - 3.48T + 11T^{2} \) |
| 13 | \( 1 - 3.78T + 13T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 - 6.98T + 19T^{2} \) |
| 23 | \( 1 - 3.91T + 23T^{2} \) |
| 29 | \( 1 - 8.03T + 29T^{2} \) |
| 31 | \( 1 - 4.13T + 31T^{2} \) |
| 37 | \( 1 + 6.15T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 9.34T + 43T^{2} \) |
| 47 | \( 1 + 5.42T + 47T^{2} \) |
| 53 | \( 1 - 5.74T + 53T^{2} \) |
| 59 | \( 1 - 1.34T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 + 5.24T + 67T^{2} \) |
| 71 | \( 1 + 9.77T + 71T^{2} \) |
| 73 | \( 1 - 5.34T + 73T^{2} \) |
| 79 | \( 1 + 9.87T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 3.17T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−8.532047656602616251293947858296, −7.946781398644506845166071762887, −7.21573018948557298229883377480, −6.68779677104027474152081723731, −5.76170194548260253225666518814, −4.49233110111055010564559656040, −3.52310751364936498744302472057, −2.90067308857436188233458670958, −1.46571876315271551500506296746, −0.890825806239082465762645577534,
0.890825806239082465762645577534, 1.46571876315271551500506296746, 2.90067308857436188233458670958, 3.52310751364936498744302472057, 4.49233110111055010564559656040, 5.76170194548260253225666518814, 6.68779677104027474152081723731, 7.21573018948557298229883377480, 7.946781398644506845166071762887, 8.532047656602616251293947858296