| L(s) = 1 | + 4.27i·5-s + (1.13 + 2.38i)7-s + 2.15·11-s + 0.274i·17-s − 4.35i·19-s − 8.71·23-s − 13.2·25-s + (−10.2 + 4.86i)35-s − 10.8·43-s + 10.2i·47-s + (−4.41 + 5.43i)49-s + 9.19i·55-s − 15.1i·61-s + 2.98i·73-s + (2.44 + 5.13i)77-s + ⋯ |
| L(s) = 1 | + 1.91i·5-s + (0.429 + 0.902i)7-s + 0.648·11-s + 0.0666i·17-s − 0.999i·19-s − 1.81·23-s − 2.65·25-s + (−1.72 + 0.821i)35-s − 1.65·43-s + 1.49i·47-s + (−0.630 + 0.776i)49-s + 1.23i·55-s − 1.94i·61-s + 0.349i·73-s + (0.278 + 0.585i)77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9400617574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9400617574\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.13 - 2.38i)T \) |
| 19 | \( 1 + 4.35iT \) |
| good | 5 | \( 1 - 4.27iT - 5T^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 0.274iT - 17T^{2} \) |
| 23 | \( 1 + 8.71T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 10.2iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15.1iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 2.98iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.563225563598359289180044552178, −7.921085743830061977397748803055, −7.19070632970462574845336615407, −6.38835848113723807933510196080, −6.12876850460984176344053991510, −5.09206850597463874524555034469, −4.06119571803000183462266258181, −3.26622630577259000769840519619, −2.49839953159653315202832061915, −1.80098587634226705777557698142,
0.24376123539604450909640035407, 1.34031154422252612277520027858, 1.88940521856649130362146197349, 3.69681731720600744952169197880, 4.11869720376379473407117227168, 4.84198793863636285278349141516, 5.56845994320938185359888370407, 6.29771277070161195996649676662, 7.34154039710492284458297199739, 8.074956729001351382337881056332
