| L(s) = 1 | + 1.29·3-s − 0.128·5-s − 1.95·7-s − 1.31·9-s − 3.98i·11-s + 3.75i·13-s − 0.167·15-s − 6.50i·17-s − 1.63i·19-s − 2.53·21-s + (4.77 + 0.420i)23-s − 4.98·25-s − 5.60·27-s − 6.60i·29-s + 0.894i·31-s + ⋯ |
| L(s) = 1 | + 0.749·3-s − 0.0576·5-s − 0.737·7-s − 0.438·9-s − 1.20i·11-s + 1.04i·13-s − 0.0432·15-s − 1.57i·17-s − 0.374i·19-s − 0.552·21-s + (0.996 + 0.0875i)23-s − 0.996·25-s − 1.07·27-s − 1.22i·29-s + 0.160i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.207652087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.207652087\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (-4.77 - 0.420i)T \) |
| good | 3 | \( 1 - 1.29T + 3T^{2} \) |
| 5 | \( 1 + 0.128T + 5T^{2} \) |
| 7 | \( 1 + 1.95T + 7T^{2} \) |
| 11 | \( 1 + 3.98iT - 11T^{2} \) |
| 13 | \( 1 - 3.75iT - 13T^{2} \) |
| 17 | \( 1 + 6.50iT - 17T^{2} \) |
| 19 | \( 1 + 1.63iT - 19T^{2} \) |
| 29 | \( 1 + 6.60iT - 29T^{2} \) |
| 31 | \( 1 - 0.894iT - 31T^{2} \) |
| 37 | \( 1 + 4.16T + 37T^{2} \) |
| 41 | \( 1 - 9.65T + 41T^{2} \) |
| 43 | \( 1 + 10.6iT - 43T^{2} \) |
| 47 | \( 1 + 9.07iT - 47T^{2} \) |
| 53 | \( 1 + 9.33T + 53T^{2} \) |
| 59 | \( 1 - 6.06T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 0.570iT - 67T^{2} \) |
| 71 | \( 1 + 0.248iT - 71T^{2} \) |
| 73 | \( 1 - 1.45T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 9.45iT - 83T^{2} \) |
| 89 | \( 1 - 11.2iT - 89T^{2} \) |
| 97 | \( 1 + 4.83iT - 97T^{2} \) |
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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
−9.240577750993543377121095922830, −8.633567850303003279105026814715, −7.70752028186467379198474819626, −6.88299881023795330575586167247, −6.03830887060115239506255312116, −5.11677778149169779728331782692, −3.86899229723569571724519806459, −3.10905519326148604433461439293, −2.27821195840234668085288699854, −0.41389757188912688035224594751,
1.64663982105011741393791319673, 2.86440580783993474448987711363, 3.53358588038717955801181963997, 4.60849028858525096968386779159, 5.75699551753479772248508104721, 6.45291422583167674805015988213, 7.63268770940305678097427179454, 8.016858227554012244459377163586, 9.052558776103954362240993462945, 9.589262529826702258346294740403
