L(s) = 1 | + 2.35·2-s + 3.56·4-s + 3.48·5-s + 7-s + 3.69·8-s + 8.21·10-s + 3.61·11-s − 1.92·13-s + 2.35·14-s + 1.58·16-s + 6.15·17-s − 4.23·19-s + 12.4·20-s + 8.53·22-s + 7.99·23-s + 7.13·25-s − 4.55·26-s + 3.56·28-s − 2.42·29-s − 1.78·31-s − 3.64·32-s + 14.5·34-s + 3.48·35-s + 5.76·37-s − 9.99·38-s + 12.8·40-s − 6.30·41-s + ⋯ |
L(s) = 1 | + 1.66·2-s + 1.78·4-s + 1.55·5-s + 0.377·7-s + 1.30·8-s + 2.59·10-s + 1.09·11-s − 0.535·13-s + 0.630·14-s + 0.397·16-s + 1.49·17-s − 0.971·19-s + 2.77·20-s + 1.81·22-s + 1.66·23-s + 1.42·25-s − 0.892·26-s + 0.674·28-s − 0.450·29-s − 0.320·31-s − 0.644·32-s + 2.48·34-s + 0.588·35-s + 0.947·37-s − 1.62·38-s + 2.03·40-s − 0.985·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(9.025158544\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.025158544\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 - T \) |
good | 2 | \( 1 - 2.35T + 2T^{2} \) |
| 5 | \( 1 - 3.48T + 5T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 + 1.92T + 13T^{2} \) |
| 17 | \( 1 - 6.15T + 17T^{2} \) |
| 19 | \( 1 + 4.23T + 19T^{2} \) |
| 23 | \( 1 - 7.99T + 23T^{2} \) |
| 29 | \( 1 + 2.42T + 29T^{2} \) |
| 31 | \( 1 + 1.78T + 31T^{2} \) |
| 37 | \( 1 - 5.76T + 37T^{2} \) |
| 41 | \( 1 + 6.30T + 41T^{2} \) |
| 43 | \( 1 + 2.65T + 43T^{2} \) |
| 47 | \( 1 + 3.09T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 + 7.38T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 8.46T + 79T^{2} \) |
| 83 | \( 1 + 8.32T + 83T^{2} \) |
| 89 | \( 1 + 5.58T + 89T^{2} \) |
| 97 | \( 1 - 3.96T + 97T^{2} \) |
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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
−7.46283335747310322219204833839, −6.76290348864831564044050674846, −6.22633554636220280044741809419, −5.66142853579046542042392069735, −5.02695451710210691836185806115, −4.52548476365320712173627195693, −3.49338973596106878346617008929, −2.88758442193579161368624614648, −1.95921661274723415617720879960, −1.33564518640688250770021025030,
1.33564518640688250770021025030, 1.95921661274723415617720879960, 2.88758442193579161368624614648, 3.49338973596106878346617008929, 4.52548476365320712173627195693, 5.02695451710210691836185806115, 5.66142853579046542042392069735, 6.22633554636220280044741809419, 6.76290348864831564044050674846, 7.46283335747310322219204833839