L(s) = 1 | + 2i·3-s − 9-s + 3·11-s + 4i·13-s + 2·19-s − 3i·23-s + 4i·27-s − 9·29-s − 8·31-s + 6i·33-s − 5i·37-s − 8·39-s + 6·41-s + 11i·43-s + 6i·47-s + ⋯ |
L(s) = 1 | + 1.15i·3-s − 0.333·9-s + 0.904·11-s + 1.10i·13-s + 0.458·19-s − 0.625i·23-s + 0.769i·27-s − 1.67·29-s − 1.43·31-s + 1.04i·33-s − 0.821i·37-s − 1.28·39-s + 0.937·41-s + 1.67i·43-s + 0.875i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.630736842\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.630736842\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 5iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 11iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 5iT - 67T^{2} \) |
| 71 | \( 1 - 15T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 7T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−8.946003870512646826768909572706, −7.82693856811984693533381047062, −7.12651668971315294136418562459, −6.36140337438827289835667728153, −5.54064799976285830487855539950, −4.75763976041528938941185958969, −3.96932067493323880111263117415, −3.67360931563469139047896521133, −2.38231624300794314603861138876, −1.32049414643531234638633372558,
0.45392276574366017408121681965, 1.47535798637462943977854971429, 2.21681233367594693491822619975, 3.40422041411587725103701471552, 4.00365918262057990918171594480, 5.40375772514510427236820358220, 5.67667741574166284614229752916, 6.80989579169979232757845307786, 7.12217062758546627375529513905, 7.86194639426013450934077220128