L(s) = 1 | − 3-s + 2.60·5-s − 3.58·7-s + 9-s − 4.72·11-s − 0.942·13-s − 2.60·15-s − 3.64·17-s − 2.69·19-s + 3.58·21-s − 0.381·23-s + 1.76·25-s − 27-s + 4.31·29-s + 0.400·31-s + 4.72·33-s − 9.31·35-s + 0.407·37-s + 0.942·39-s + 0.759·41-s + 4.47·43-s + 2.60·45-s + 13.2·47-s + 5.82·49-s + 3.64·51-s + 4.41·53-s − 12.2·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.16·5-s − 1.35·7-s + 0.333·9-s − 1.42·11-s − 0.261·13-s − 0.671·15-s − 0.884·17-s − 0.617·19-s + 0.781·21-s − 0.0795·23-s + 0.353·25-s − 0.192·27-s + 0.800·29-s + 0.0719·31-s + 0.822·33-s − 1.57·35-s + 0.0669·37-s + 0.150·39-s + 0.118·41-s + 0.682·43-s + 0.387·45-s + 1.92·47-s + 0.832·49-s + 0.510·51-s + 0.606·53-s − 1.65·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.073264325\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.073264325\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 - 2.60T + 5T^{2} \) |
| 7 | \( 1 + 3.58T + 7T^{2} \) |
| 11 | \( 1 + 4.72T + 11T^{2} \) |
| 13 | \( 1 + 0.942T + 13T^{2} \) |
| 17 | \( 1 + 3.64T + 17T^{2} \) |
| 19 | \( 1 + 2.69T + 19T^{2} \) |
| 23 | \( 1 + 0.381T + 23T^{2} \) |
| 29 | \( 1 - 4.31T + 29T^{2} \) |
| 31 | \( 1 - 0.400T + 31T^{2} \) |
| 37 | \( 1 - 0.407T + 37T^{2} \) |
| 41 | \( 1 - 0.759T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 - 4.41T + 53T^{2} \) |
| 59 | \( 1 + 5.47T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 4.38T + 67T^{2} \) |
| 71 | \( 1 + 1.17T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 8.57T + 79T^{2} \) |
| 83 | \( 1 - 0.551T + 83T^{2} \) |
| 89 | \( 1 - 0.764T + 89T^{2} \) |
| 97 | \( 1 + 4.11T + 97T^{2} \) |
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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
−8.573283894286668598959312097650, −7.56745721436107790038168166343, −6.74436698956515127736342224143, −6.21385980173963020367322005980, −5.60218156673484989879333536660, −4.89223598143472248548777122997, −3.88482054372392999043161255725, −2.65996445227458727209422397086, −2.22196148123192659785053069413, −0.57545577491895072579843430623,
0.57545577491895072579843430623, 2.22196148123192659785053069413, 2.65996445227458727209422397086, 3.88482054372392999043161255725, 4.89223598143472248548777122997, 5.60218156673484989879333536660, 6.21385980173963020367322005980, 6.74436698956515127736342224143, 7.56745721436107790038168166343, 8.573283894286668598959312097650