| L(s) = 1 | + 5-s − 0.807i·7-s − 2.90·11-s + 2.37·13-s − 6.47·17-s + 3.20i·19-s + (2.82 + 3.87i)23-s + 25-s − 7.84i·29-s − 3.15·31-s − 0.807i·35-s − 7.68i·37-s − 0.186i·41-s − 2.61i·43-s − 10.5i·47-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.305i·7-s − 0.876·11-s + 0.659·13-s − 1.57·17-s + 0.735i·19-s + (0.589 + 0.807i)23-s + 0.200·25-s − 1.45i·29-s − 0.567·31-s − 0.136i·35-s − 1.26i·37-s − 0.0291i·41-s − 0.398i·43-s − 1.54i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.151717428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.151717428\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + (-2.82 - 3.87i)T \) |
| good | 7 | \( 1 + 0.807iT - 7T^{2} \) |
| 11 | \( 1 + 2.90T + 11T^{2} \) |
| 13 | \( 1 - 2.37T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 3.20iT - 19T^{2} \) |
| 29 | \( 1 + 7.84iT - 29T^{2} \) |
| 31 | \( 1 + 3.15T + 31T^{2} \) |
| 37 | \( 1 + 7.68iT - 37T^{2} \) |
| 41 | \( 1 + 0.186iT - 41T^{2} \) |
| 43 | \( 1 + 2.61iT - 43T^{2} \) |
| 47 | \( 1 + 10.5iT - 47T^{2} \) |
| 53 | \( 1 - 5.14T + 53T^{2} \) |
| 59 | \( 1 + 0.0710iT - 59T^{2} \) |
| 61 | \( 1 + 5.76iT - 61T^{2} \) |
| 67 | \( 1 + 2.04iT - 67T^{2} \) |
| 71 | \( 1 + 4.39iT - 71T^{2} \) |
| 73 | \( 1 + 4.19T + 73T^{2} \) |
| 79 | \( 1 - 0.308iT - 79T^{2} \) |
| 83 | \( 1 + 0.614T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 2.32iT - 97T^{2} \) |
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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
−8.257764546906847852397228828104, −7.41146374857749365915635916141, −6.79652698139401829697521328657, −5.86263001753472089162994634406, −5.38880670573580490110874007767, −4.33350547662291242067904759643, −3.66271793305323275792989933656, −2.51664864180988204671452784041, −1.76922616038273943056166374769, −0.32425892216222323402223769865,
1.21966513854636171808769614731, 2.41944430059465971046216061137, 2.96436106972113394117980722254, 4.22398990473043385567313693988, 4.94237757219808791883784132446, 5.63879972178993638827813545130, 6.54614410825035988986620323220, 6.99597656143557058603878565475, 8.005458999198367458003496884896, 8.894756307780826521850036193728
