| L(s) = 1 | + 0.618·2-s + 0.381·3-s − 1.61·4-s + 0.236·6-s − 3.85·7-s − 2.23·8-s − 2.85·9-s − 0.618·12-s − 1.76·13-s − 2.38·14-s + 1.85·16-s − 1.61·17-s − 1.76·18-s − 6.70·19-s − 1.47·21-s + 7.09·23-s − 0.854·24-s − 1.09·26-s − 2.23·27-s + 6.23·28-s + 3.61·29-s − 3·31-s + 5.61·32-s − 1.00·34-s + 4.61·36-s + 5.76·37-s − 4.14·38-s + ⋯ |
| L(s) = 1 | + 0.437·2-s + 0.220·3-s − 0.809·4-s + 0.0963·6-s − 1.45·7-s − 0.790·8-s − 0.951·9-s − 0.178·12-s − 0.489·13-s − 0.636·14-s + 0.463·16-s − 0.392·17-s − 0.415·18-s − 1.53·19-s − 0.321·21-s + 1.47·23-s − 0.174·24-s − 0.213·26-s − 0.430·27-s + 1.17·28-s + 0.671·29-s − 0.538·31-s + 0.993·32-s − 0.171·34-s + 0.769·36-s + 0.947·37-s − 0.672·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8416813393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8416813393\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 3 | \( 1 - 0.381T + 3T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 - 7.09T + 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 5.76T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 5.94T + 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 - 9.47T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 0.854T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 + 0.618T + 97T^{2} \) |
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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
−8.827885907428486798157694586701, −8.201446986708997187654314901168, −7.04122266132773873598011453079, −6.32657147907866032854651657737, −5.70333016856872046537591203535, −4.77824801031410966214136859996, −3.97789620170783798226481701237, −3.10436538494088498147198008226, −2.51347707589512151158421849282, −0.48814740112594634628912402620,
0.48814740112594634628912402620, 2.51347707589512151158421849282, 3.10436538494088498147198008226, 3.97789620170783798226481701237, 4.77824801031410966214136859996, 5.70333016856872046537591203535, 6.32657147907866032854651657737, 7.04122266132773873598011453079, 8.201446986708997187654314901168, 8.827885907428486798157694586701
