L(s) = 1 | − 2-s − 1.26·3-s + 4-s + 1.05·5-s + 1.26·6-s + 0.0899·7-s − 8-s − 1.38·9-s − 1.05·10-s + 4.74·11-s − 1.26·12-s + 2.51·13-s − 0.0899·14-s − 1.33·15-s + 16-s − 7.81·17-s + 1.38·18-s − 4.19·19-s + 1.05·20-s − 0.114·21-s − 4.74·22-s − 3.87·23-s + 1.26·24-s − 3.89·25-s − 2.51·26-s + 5.57·27-s + 0.0899·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.732·3-s + 0.5·4-s + 0.470·5-s + 0.518·6-s + 0.0340·7-s − 0.353·8-s − 0.463·9-s − 0.332·10-s + 1.42·11-s − 0.366·12-s + 0.696·13-s − 0.0240·14-s − 0.344·15-s + 0.250·16-s − 1.89·17-s + 0.327·18-s − 0.963·19-s + 0.235·20-s − 0.0249·21-s − 1.01·22-s − 0.807·23-s + 0.259·24-s − 0.778·25-s − 0.492·26-s + 1.07·27-s + 0.0170·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9684829668\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9684829668\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
good | 3 | \( 1 + 1.26T + 3T^{2} \) |
| 5 | \( 1 - 1.05T + 5T^{2} \) |
| 7 | \( 1 - 0.0899T + 7T^{2} \) |
| 11 | \( 1 - 4.74T + 11T^{2} \) |
| 13 | \( 1 - 2.51T + 13T^{2} \) |
| 17 | \( 1 + 7.81T + 17T^{2} \) |
| 19 | \( 1 + 4.19T + 19T^{2} \) |
| 23 | \( 1 + 3.87T + 23T^{2} \) |
| 29 | \( 1 - 1.30T + 29T^{2} \) |
| 31 | \( 1 - 6.51T + 31T^{2} \) |
| 37 | \( 1 - 8.50T + 37T^{2} \) |
| 41 | \( 1 - 7.45T + 41T^{2} \) |
| 43 | \( 1 - 7.56T + 43T^{2} \) |
| 47 | \( 1 - 3.50T + 47T^{2} \) |
| 53 | \( 1 + 2.44T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 4.35T + 67T^{2} \) |
| 71 | \( 1 - 16.6T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 9.33T + 79T^{2} \) |
| 83 | \( 1 - 3.61T + 83T^{2} \) |
| 89 | \( 1 + 4.48T + 89T^{2} \) |
| 97 | \( 1 - 7.97T + 97T^{2} \) |
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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
−8.531860868506985127612883611814, −7.88456801828356869634503949054, −6.65404452411104022818357747566, −6.25436857404000844979427338309, −5.97480809261255239214781024782, −4.58769624199704884423951981403, −4.02390554061708598920675196259, −2.64814736336672900225724567103, −1.80188587774225353127256899351, −0.65004558391671695292358896747,
0.65004558391671695292358896747, 1.80188587774225353127256899351, 2.64814736336672900225724567103, 4.02390554061708598920675196259, 4.58769624199704884423951981403, 5.97480809261255239214781024782, 6.25436857404000844979427338309, 6.65404452411104022818357747566, 7.88456801828356869634503949054, 8.531860868506985127612883611814