| L(s) = 1 | + 3-s + 0.381·7-s + 9-s + 11-s + 2.23·13-s + 0.381·17-s − 2.23·19-s + 0.381·21-s + 6.23·23-s + 27-s + 1.76·29-s + 2.09·31-s + 33-s − 4.70·37-s + 2.23·39-s + 3.61·41-s + 7.09·43-s − 0.708·47-s − 6.85·49-s + 0.381·51-s + 11.0·53-s − 2.23·57-s + 2.09·59-s − 4.38·61-s + 0.381·63-s + 10.4·67-s + 6.23·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.144·7-s + 0.333·9-s + 0.301·11-s + 0.620·13-s + 0.0926·17-s − 0.512·19-s + 0.0833·21-s + 1.30·23-s + 0.192·27-s + 0.327·29-s + 0.375·31-s + 0.174·33-s − 0.774·37-s + 0.358·39-s + 0.565·41-s + 1.08·43-s − 0.103·47-s − 0.979·49-s + 0.0534·51-s + 1.52·53-s − 0.296·57-s + 0.272·59-s − 0.561·61-s + 0.0481·63-s + 1.27·67-s + 0.750·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.278724837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.278724837\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 0.381T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 2.23T + 13T^{2} \) |
| 17 | \( 1 - 0.381T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 - 6.23T + 23T^{2} \) |
| 29 | \( 1 - 1.76T + 29T^{2} \) |
| 31 | \( 1 - 2.09T + 31T^{2} \) |
| 37 | \( 1 + 4.70T + 37T^{2} \) |
| 41 | \( 1 - 3.61T + 41T^{2} \) |
| 43 | \( 1 - 7.09T + 43T^{2} \) |
| 47 | \( 1 + 0.708T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 2.09T + 59T^{2} \) |
| 61 | \( 1 + 4.38T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 5.85T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 - 3.14T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−9.286282884567675136164028380632, −8.739588614276958436270263529478, −7.996870039505144835594481102654, −7.08637774188322291803912120049, −6.34957509370589126845673278409, −5.28918260568943743362276744793, −4.31210506948065597465865562603, −3.43165328376635020530748790828, −2.40193315771154404133280514444, −1.13142442332657512950501626148,
1.13142442332657512950501626148, 2.40193315771154404133280514444, 3.43165328376635020530748790828, 4.31210506948065597465865562603, 5.28918260568943743362276744793, 6.34957509370589126845673278409, 7.08637774188322291803912120049, 7.996870039505144835594481102654, 8.739588614276958436270263529478, 9.286282884567675136164028380632
