L(s) = 1 | − 2-s + 4-s − 4.32·5-s + 2.79·7-s − 8-s + 4.32·10-s + 2.77·11-s − 4.55·13-s − 2.79·14-s + 16-s + 7.65·17-s − 0.819·19-s − 4.32·20-s − 2.77·22-s + 3.06·23-s + 13.7·25-s + 4.55·26-s + 2.79·28-s − 2.92·29-s − 1.60·31-s − 32-s − 7.65·34-s − 12.1·35-s − 4.99·37-s + 0.819·38-s + 4.32·40-s − 4.19·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.93·5-s + 1.05·7-s − 0.353·8-s + 1.36·10-s + 0.835·11-s − 1.26·13-s − 0.747·14-s + 0.250·16-s + 1.85·17-s − 0.187·19-s − 0.967·20-s − 0.590·22-s + 0.638·23-s + 2.74·25-s + 0.892·26-s + 0.528·28-s − 0.543·29-s − 0.287·31-s − 0.176·32-s − 1.31·34-s − 2.04·35-s − 0.820·37-s + 0.132·38-s + 0.684·40-s − 0.654·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9409066920\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9409066920\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 223 | \( 1 + T \) |
good | 5 | \( 1 + 4.32T + 5T^{2} \) |
| 7 | \( 1 - 2.79T + 7T^{2} \) |
| 11 | \( 1 - 2.77T + 11T^{2} \) |
| 13 | \( 1 + 4.55T + 13T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 19 | \( 1 + 0.819T + 19T^{2} \) |
| 23 | \( 1 - 3.06T + 23T^{2} \) |
| 29 | \( 1 + 2.92T + 29T^{2} \) |
| 31 | \( 1 + 1.60T + 31T^{2} \) |
| 37 | \( 1 + 4.99T + 37T^{2} \) |
| 41 | \( 1 + 4.19T + 41T^{2} \) |
| 43 | \( 1 - 6.95T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 - 2.54T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 1.42T + 61T^{2} \) |
| 67 | \( 1 - 5.12T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 9.56T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 6.23T + 83T^{2} \) |
| 89 | \( 1 - 3.15T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−8.268644236883803349553361477456, −7.68646826565534599755193960359, −7.42008239361119295203319582626, −6.59495677384890405657269045299, −5.25496710716203657877551380090, −4.69887302947875826716939948280, −3.72396810734122748735942188883, −3.09485880205996195714066071804, −1.68133347259820368418004461284, −0.63980567483382581177600725725,
0.63980567483382581177600725725, 1.68133347259820368418004461284, 3.09485880205996195714066071804, 3.72396810734122748735942188883, 4.69887302947875826716939948280, 5.25496710716203657877551380090, 6.59495677384890405657269045299, 7.42008239361119295203319582626, 7.68646826565534599755193960359, 8.268644236883803349553361477456