L(s) = 1 | − 5-s − 5.70·11-s − 6.70·13-s − 3.44·17-s + 1.25·19-s + 7.96·23-s + 25-s + 7.18·29-s − 31-s − 8.96·37-s − 10.2·41-s − 4.44·43-s − 1.48·47-s − 6.89·53-s + 5.70·55-s + 0.293·59-s − 0.259·61-s + 6.70·65-s − 6.70·67-s + 11.7·71-s + 0.966·73-s − 11.6·79-s + 14.8·83-s + 3.44·85-s + 12.6·89-s − 1.25·95-s − 13.1·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.72·11-s − 1.86·13-s − 0.836·17-s + 0.288·19-s + 1.66·23-s + 0.200·25-s + 1.33·29-s − 0.179·31-s − 1.47·37-s − 1.59·41-s − 0.678·43-s − 0.216·47-s − 0.947·53-s + 0.769·55-s + 0.0381·59-s − 0.0332·61-s + 0.831·65-s − 0.819·67-s + 1.38·71-s + 0.113·73-s − 1.30·79-s + 1.63·83-s + 0.373·85-s + 1.33·89-s − 0.129·95-s − 1.33·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7028851280\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7028851280\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 5.70T + 11T^{2} \) |
| 13 | \( 1 + 6.70T + 13T^{2} \) |
| 17 | \( 1 + 3.44T + 17T^{2} \) |
| 19 | \( 1 - 1.25T + 19T^{2} \) |
| 23 | \( 1 - 7.96T + 23T^{2} \) |
| 29 | \( 1 - 7.18T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 8.96T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 4.44T + 43T^{2} \) |
| 47 | \( 1 + 1.48T + 47T^{2} \) |
| 53 | \( 1 + 6.89T + 53T^{2} \) |
| 59 | \( 1 - 0.293T + 59T^{2} \) |
| 61 | \( 1 + 0.259T + 61T^{2} \) |
| 67 | \( 1 + 6.70T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 0.966T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−7.81206286715586738776316485382, −6.94348249264826125528501009759, −6.77088584990498875978858640366, −5.27992991243703387087997354217, −5.08838194564283236594058058263, −4.49464208910651960342705194981, −3.18508122873793967674685589087, −2.79190048783959710160481967504, −1.88293442085013186933000536540, −0.38217644204947088472056642918,
0.38217644204947088472056642918, 1.88293442085013186933000536540, 2.79190048783959710160481967504, 3.18508122873793967674685589087, 4.49464208910651960342705194981, 5.08838194564283236594058058263, 5.27992991243703387087997354217, 6.77088584990498875978858640366, 6.94348249264826125528501009759, 7.81206286715586738776316485382