L(s) = 1 | + 1.75·2-s − 3.22·3-s + 1.08·4-s − 1.92·5-s − 5.66·6-s − 1.60·8-s + 7.37·9-s − 3.37·10-s − 2.88·11-s − 3.50·12-s − 6.75·13-s + 6.19·15-s − 4.99·16-s + 5.66·17-s + 12.9·18-s − 1.95·19-s − 2.09·20-s − 5.07·22-s − 0.0347·23-s + 5.15·24-s − 1.30·25-s − 11.8·26-s − 14.0·27-s − 5.94·29-s + 10.8·30-s + 4.61·31-s − 5.56·32-s + ⋯ |
L(s) = 1 | + 1.24·2-s − 1.85·3-s + 0.544·4-s − 0.859·5-s − 2.31·6-s − 0.566·8-s + 2.45·9-s − 1.06·10-s − 0.870·11-s − 1.01·12-s − 1.87·13-s + 1.59·15-s − 1.24·16-s + 1.37·17-s + 3.05·18-s − 0.448·19-s − 0.467·20-s − 1.08·22-s − 0.00724·23-s + 1.05·24-s − 0.261·25-s − 2.32·26-s − 2.71·27-s − 1.10·29-s + 1.98·30-s + 0.829·31-s − 0.984·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6998168333\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6998168333\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 - 1.75T + 2T^{2} \) |
| 3 | \( 1 + 3.22T + 3T^{2} \) |
| 5 | \( 1 + 1.92T + 5T^{2} \) |
| 11 | \( 1 + 2.88T + 11T^{2} \) |
| 13 | \( 1 + 6.75T + 13T^{2} \) |
| 17 | \( 1 - 5.66T + 17T^{2} \) |
| 19 | \( 1 + 1.95T + 19T^{2} \) |
| 23 | \( 1 + 0.0347T + 23T^{2} \) |
| 29 | \( 1 + 5.94T + 29T^{2} \) |
| 31 | \( 1 - 4.61T + 31T^{2} \) |
| 37 | \( 1 - 6.06T + 37T^{2} \) |
| 43 | \( 1 - 2.48T + 43T^{2} \) |
| 47 | \( 1 - 8.38T + 47T^{2} \) |
| 53 | \( 1 + 5.94T + 53T^{2} \) |
| 59 | \( 1 - 5.99T + 59T^{2} \) |
| 61 | \( 1 - 4.46T + 61T^{2} \) |
| 67 | \( 1 - 8.16T + 67T^{2} \) |
| 71 | \( 1 - 1.94T + 71T^{2} \) |
| 73 | \( 1 + 5.10T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 - 5.74T + 83T^{2} \) |
| 89 | \( 1 + 0.807T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−9.532156379765426435981418337784, −7.87548773689163787707572866600, −7.38904729402604167715797579780, −6.49682316398642276698827103127, −5.60070472047979844486218892477, −5.18652309388947384244641704228, −4.51262579666437185480247240191, −3.76398478037850903125072754903, −2.45690889755236209470195114137, −0.48236090735567782070871392127,
0.48236090735567782070871392127, 2.45690889755236209470195114137, 3.76398478037850903125072754903, 4.51262579666437185480247240191, 5.18652309388947384244641704228, 5.60070472047979844486218892477, 6.49682316398642276698827103127, 7.38904729402604167715797579780, 7.87548773689163787707572866600, 9.532156379765426435981418337784