| L(s) = 1 | + (−2.09 + 0.770i)5-s + 2.34i·7-s + 4.15·11-s − 5.10i·13-s + 0.0253i·17-s + 0.536·19-s − i·23-s + (3.81 − 3.23i)25-s − 9.72·29-s − 8.15·31-s + (−1.80 − 4.91i)35-s + 2.50i·37-s + 4.20·41-s − 11.5i·43-s − 1.22i·47-s + ⋯ |
| L(s) = 1 | + (−0.938 + 0.344i)5-s + 0.884i·7-s + 1.25·11-s − 1.41i·13-s + 0.00615i·17-s + 0.123·19-s − 0.208i·23-s + (0.762 − 0.647i)25-s − 1.80·29-s − 1.46·31-s + (−0.304 − 0.830i)35-s + 0.411i·37-s + 0.656·41-s − 1.75i·43-s − 0.178i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.100515434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.100515434\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.09 - 0.770i)T \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 - 2.34iT - 7T^{2} \) |
| 11 | \( 1 - 4.15T + 11T^{2} \) |
| 13 | \( 1 + 5.10iT - 13T^{2} \) |
| 17 | \( 1 - 0.0253iT - 17T^{2} \) |
| 19 | \( 1 - 0.536T + 19T^{2} \) |
| 29 | \( 1 + 9.72T + 29T^{2} \) |
| 31 | \( 1 + 8.15T + 31T^{2} \) |
| 37 | \( 1 - 2.50iT - 37T^{2} \) |
| 41 | \( 1 - 4.20T + 41T^{2} \) |
| 43 | \( 1 + 11.5iT - 43T^{2} \) |
| 47 | \( 1 + 1.22iT - 47T^{2} \) |
| 53 | \( 1 - 14.3iT - 53T^{2} \) |
| 59 | \( 1 + 4.95T + 59T^{2} \) |
| 61 | \( 1 - 8.57T + 61T^{2} \) |
| 67 | \( 1 + 8.41iT - 67T^{2} \) |
| 71 | \( 1 - 1.22T + 71T^{2} \) |
| 73 | \( 1 + 3.30iT - 73T^{2} \) |
| 79 | \( 1 + 9.41T + 79T^{2} \) |
| 83 | \( 1 + 3.55iT - 83T^{2} \) |
| 89 | \( 1 + 6.97T + 89T^{2} \) |
| 97 | \( 1 + 9.13iT - 97T^{2} \) |
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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
−8.315414252267885552192531072954, −7.45828166448214620039459074908, −7.02867776236675193361558910163, −5.87765514053895940765403695784, −5.52851217459022674415558239248, −4.33910745336400847534449951161, −3.61437479137487710910735329328, −2.92983646307448959227068010493, −1.78506628823766180493521462823, −0.36643538977937497435795510537,
1.03070857034238318645672535117, 1.94545780036075508893206728226, 3.55694807088511784479956485912, 3.93402190859516422333755741538, 4.54203001064391397259051264512, 5.56869786859317807738968171374, 6.59921404399062464115180104405, 7.16235787398505726350568906720, 7.66275027956549846816014742218, 8.626487724889121777738866161565
