L(s) = 1 | + 1.14·2-s − 2.15·3-s − 0.679·4-s + 3.87·5-s − 2.47·6-s − 3.07·8-s + 1.63·9-s + 4.45·10-s + 1.46·12-s + 4.09·13-s − 8.34·15-s − 2.17·16-s − 0.824·17-s + 1.87·18-s + 4.50·19-s − 2.63·20-s + 4.86·23-s + 6.62·24-s + 10.0·25-s + 4.70·26-s + 2.93·27-s − 7.79·29-s − 9.58·30-s − 3.82·31-s + 3.65·32-s − 0.947·34-s − 1.11·36-s + ⋯ |
L(s) = 1 | + 0.812·2-s − 1.24·3-s − 0.339·4-s + 1.73·5-s − 1.00·6-s − 1.08·8-s + 0.545·9-s + 1.40·10-s + 0.422·12-s + 1.13·13-s − 2.15·15-s − 0.544·16-s − 0.200·17-s + 0.442·18-s + 1.03·19-s − 0.589·20-s + 1.01·23-s + 1.35·24-s + 2.00·25-s + 0.923·26-s + 0.565·27-s − 1.44·29-s − 1.75·30-s − 0.687·31-s + 0.646·32-s − 0.162·34-s − 0.185·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.314927267\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.314927267\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.14T + 2T^{2} \) |
| 3 | \( 1 + 2.15T + 3T^{2} \) |
| 5 | \( 1 - 3.87T + 5T^{2} \) |
| 13 | \( 1 - 4.09T + 13T^{2} \) |
| 17 | \( 1 + 0.824T + 17T^{2} \) |
| 19 | \( 1 - 4.50T + 19T^{2} \) |
| 23 | \( 1 - 4.86T + 23T^{2} \) |
| 29 | \( 1 + 7.79T + 29T^{2} \) |
| 31 | \( 1 + 3.82T + 31T^{2} \) |
| 37 | \( 1 - 8.11T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 1.86T + 43T^{2} \) |
| 47 | \( 1 + 7.69T + 47T^{2} \) |
| 53 | \( 1 + 3.93T + 53T^{2} \) |
| 59 | \( 1 - 9.45T + 59T^{2} \) |
| 61 | \( 1 + 8.40T + 61T^{2} \) |
| 67 | \( 1 - 9.45T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 6.19T + 73T^{2} \) |
| 79 | \( 1 + 0.868T + 79T^{2} \) |
| 83 | \( 1 - 8.19T + 83T^{2} \) |
| 89 | \( 1 - 3.87T + 89T^{2} \) |
| 97 | \( 1 - 2.10T + 97T^{2} \) |
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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
−8.134162919979900925683888384079, −6.80814033792700275521790146841, −6.40827808246972935885577073319, −5.71295091707496586116142192539, −5.32736787666797505964693157068, −4.89772980631523550224298357270, −3.74677774569601849200766152718, −2.95148656354468153603368898245, −1.76817577226686985236809729018, −0.78705560506284664521544283832,
0.78705560506284664521544283832, 1.76817577226686985236809729018, 2.95148656354468153603368898245, 3.74677774569601849200766152718, 4.89772980631523550224298357270, 5.32736787666797505964693157068, 5.71295091707496586116142192539, 6.40827808246972935885577073319, 6.80814033792700275521790146841, 8.134162919979900925683888384079