L(s) = 1 | + 2.18·2-s + 1.16·3-s + 2.78·4-s + 0.568·5-s + 2.54·6-s + 0.610·7-s + 1.71·8-s − 1.64·9-s + 1.24·10-s − 2.93·11-s + 3.24·12-s − 0.274·13-s + 1.33·14-s + 0.662·15-s − 1.80·16-s + 0.122·17-s − 3.59·18-s + 1.54·19-s + 1.58·20-s + 0.711·21-s − 6.43·22-s + 0.331·23-s + 2.00·24-s − 4.67·25-s − 0.599·26-s − 5.40·27-s + 1.70·28-s + ⋯ |
L(s) = 1 | + 1.54·2-s + 0.672·3-s + 1.39·4-s + 0.254·5-s + 1.04·6-s + 0.230·7-s + 0.608·8-s − 0.547·9-s + 0.393·10-s − 0.886·11-s + 0.937·12-s − 0.0760·13-s + 0.357·14-s + 0.171·15-s − 0.452·16-s + 0.0297·17-s − 0.846·18-s + 0.353·19-s + 0.354·20-s + 0.155·21-s − 1.37·22-s + 0.0692·23-s + 0.409·24-s − 0.935·25-s − 0.117·26-s − 1.04·27-s + 0.321·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.379669333\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.379669333\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 349 | \( 1 - T \) |
good | 2 | \( 1 - 2.18T + 2T^{2} \) |
| 3 | \( 1 - 1.16T + 3T^{2} \) |
| 5 | \( 1 - 0.568T + 5T^{2} \) |
| 7 | \( 1 - 0.610T + 7T^{2} \) |
| 11 | \( 1 + 2.93T + 11T^{2} \) |
| 13 | \( 1 + 0.274T + 13T^{2} \) |
| 17 | \( 1 - 0.122T + 17T^{2} \) |
| 19 | \( 1 - 1.54T + 19T^{2} \) |
| 23 | \( 1 - 0.331T + 23T^{2} \) |
| 29 | \( 1 - 8.59T + 29T^{2} \) |
| 31 | \( 1 - 5.79T + 31T^{2} \) |
| 37 | \( 1 + 3.66T + 37T^{2} \) |
| 41 | \( 1 - 8.17T + 41T^{2} \) |
| 43 | \( 1 - 1.77T + 43T^{2} \) |
| 47 | \( 1 - 1.23T + 47T^{2} \) |
| 53 | \( 1 - 3.59T + 53T^{2} \) |
| 59 | \( 1 + 5.50T + 59T^{2} \) |
| 61 | \( 1 + 4.74T + 61T^{2} \) |
| 67 | \( 1 - 5.95T + 67T^{2} \) |
| 71 | \( 1 + 1.76T + 71T^{2} \) |
| 73 | \( 1 + 1.64T + 73T^{2} \) |
| 79 | \( 1 - 4.55T + 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 + 4.83T + 89T^{2} \) |
| 97 | \( 1 - 0.739T + 97T^{2} \) |
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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
−11.79516547252816319145083946205, −10.85538073968581840445824045578, −9.710022478854163451593741832517, −8.543771065996849862504739585980, −7.62458150506233053767024303014, −6.31126317562739023549702392300, −5.43961528860737172928476306259, −4.47939518061021882317019975836, −3.19675279107537193837739699246, −2.38375320953741279526994077965,
2.38375320953741279526994077965, 3.19675279107537193837739699246, 4.47939518061021882317019975836, 5.43961528860737172928476306259, 6.31126317562739023549702392300, 7.62458150506233053767024303014, 8.543771065996849862504739585980, 9.710022478854163451593741832517, 10.85538073968581840445824045578, 11.79516547252816319145083946205