| L(s) = 1 | − 3-s − 0.0883·7-s + 9-s + 2.26·11-s + 2.65·13-s + 2.08·17-s + 1.76·19-s + 0.0883·21-s + 4.74·23-s − 27-s + 3.70·29-s − 4.10·31-s − 2.26·33-s + 7.11·37-s − 2.65·39-s − 6.58·41-s + 1.79·43-s + 10.1·47-s − 6.99·49-s − 2.08·51-s − 0.961·53-s − 1.76·57-s − 8.97·59-s + 9.46·61-s − 0.0883·63-s + 13.9·67-s − 4.74·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.0333·7-s + 0.333·9-s + 0.684·11-s + 0.735·13-s + 0.506·17-s + 0.403·19-s + 0.0192·21-s + 0.988·23-s − 0.192·27-s + 0.688·29-s − 0.737·31-s − 0.395·33-s + 1.17·37-s − 0.424·39-s − 1.02·41-s + 0.273·43-s + 1.47·47-s − 0.998·49-s − 0.292·51-s − 0.132·53-s − 0.233·57-s − 1.16·59-s + 1.21·61-s − 0.0111·63-s + 1.69·67-s − 0.570·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.956486096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.956486096\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.0883T + 7T^{2} \) |
| 11 | \( 1 - 2.26T + 11T^{2} \) |
| 13 | \( 1 - 2.65T + 13T^{2} \) |
| 17 | \( 1 - 2.08T + 17T^{2} \) |
| 19 | \( 1 - 1.76T + 19T^{2} \) |
| 23 | \( 1 - 4.74T + 23T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 + 4.10T + 31T^{2} \) |
| 37 | \( 1 - 7.11T + 37T^{2} \) |
| 41 | \( 1 + 6.58T + 41T^{2} \) |
| 43 | \( 1 - 1.79T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + 0.961T + 53T^{2} \) |
| 59 | \( 1 + 8.97T + 59T^{2} \) |
| 61 | \( 1 - 9.46T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 - 6.14T + 71T^{2} \) |
| 73 | \( 1 + 3.15T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−7.897080751709875210354214137074, −6.98466785809059179775022603208, −6.58529198101542543327219048297, −5.74298952122911073710161889385, −5.21188348653748421085015619814, −4.31111104602705126973556160996, −3.63022703798284811522695058622, −2.77508852584323530852142712412, −1.53639521437431482737279223212, −0.78823928781594038425244122620,
0.78823928781594038425244122620, 1.53639521437431482737279223212, 2.77508852584323530852142712412, 3.63022703798284811522695058622, 4.31111104602705126973556160996, 5.21188348653748421085015619814, 5.74298952122911073710161889385, 6.58529198101542543327219048297, 6.98466785809059179775022603208, 7.897080751709875210354214137074
