| L(s) = 1 | + 0.840·2-s − 3-s − 1.29·4-s + 2.01·5-s − 0.840·6-s + 4.87·7-s − 2.76·8-s + 9-s + 1.69·10-s − 11-s + 1.29·12-s + 4.10·14-s − 2.01·15-s + 0.256·16-s + 2.60·17-s + 0.840·18-s + 4.98·19-s − 2.60·20-s − 4.87·21-s − 0.840·22-s + 4.29·23-s + 2.76·24-s − 0.952·25-s − 27-s − 6.30·28-s − 3.22·29-s − 1.69·30-s + ⋯ |
| L(s) = 1 | + 0.594·2-s − 0.577·3-s − 0.646·4-s + 0.899·5-s − 0.343·6-s + 1.84·7-s − 0.979·8-s + 0.333·9-s + 0.535·10-s − 0.301·11-s + 0.373·12-s + 1.09·14-s − 0.519·15-s + 0.0641·16-s + 0.630·17-s + 0.198·18-s + 1.14·19-s − 0.581·20-s − 1.06·21-s − 0.179·22-s + 0.895·23-s + 0.565·24-s − 0.190·25-s − 0.192·27-s − 1.19·28-s − 0.598·29-s − 0.308·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.854527064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.854527064\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.840T + 2T^{2} \) |
| 5 | \( 1 - 2.01T + 5T^{2} \) |
| 7 | \( 1 - 4.87T + 7T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 19 | \( 1 - 4.98T + 19T^{2} \) |
| 23 | \( 1 - 4.29T + 23T^{2} \) |
| 29 | \( 1 + 3.22T + 29T^{2} \) |
| 31 | \( 1 - 1.26T + 31T^{2} \) |
| 37 | \( 1 - 7.08T + 37T^{2} \) |
| 41 | \( 1 - 5.87T + 41T^{2} \) |
| 43 | \( 1 + 8.59T + 43T^{2} \) |
| 47 | \( 1 + 7.80T + 47T^{2} \) |
| 53 | \( 1 + 3.45T + 53T^{2} \) |
| 59 | \( 1 - 4.95T + 59T^{2} \) |
| 61 | \( 1 + 9.00T + 61T^{2} \) |
| 67 | \( 1 - 6.12T + 67T^{2} \) |
| 71 | \( 1 + 2.00T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 + 6.28T + 79T^{2} \) |
| 83 | \( 1 - 9.11T + 83T^{2} \) |
| 89 | \( 1 + 1.64T + 89T^{2} \) |
| 97 | \( 1 - 6.83T + 97T^{2} \) |
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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
−7.984446679809403583118973461202, −7.59670967251415515942965062588, −6.43798228083017296673126537186, −5.66531849289039534038687840998, −5.15183173706499352476460830571, −4.84844468424650777817974483951, −3.91696724642450001354230434489, −2.86993656047631429604902067360, −1.75085692329031773653640727720, −0.925446198333616166868660966569,
0.925446198333616166868660966569, 1.75085692329031773653640727720, 2.86993656047631429604902067360, 3.91696724642450001354230434489, 4.84844468424650777817974483951, 5.15183173706499352476460830571, 5.66531849289039534038687840998, 6.43798228083017296673126537186, 7.59670967251415515942965062588, 7.984446679809403583118973461202
