L(s) = 1 | + (2.57 − 0.595i)7-s − 3.74i·11-s + 3.36i·13-s + 0.841·17-s − 5.59i·19-s + 2.35i·23-s − 1.41i·29-s + 8.66i·31-s + 5.15·37-s − 5.74·41-s + 3.32·43-s + 6.43·47-s + (6.29 − 3.06i)49-s − 9.64i·53-s + 12.2·59-s + ⋯ |
L(s) = 1 | + (0.974 − 0.224i)7-s − 1.12i·11-s + 0.931i·13-s + 0.204·17-s − 1.28i·19-s + 0.490i·23-s − 0.262i·29-s + 1.55i·31-s + 0.847·37-s − 0.896·41-s + 0.507·43-s + 0.938·47-s + (0.898 − 0.438i)49-s − 1.32i·53-s + 1.59·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.264334816\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.264334816\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-2.57 + 0.595i)T \) |
good | 11 | \( 1 + 3.74iT - 11T^{2} \) |
| 13 | \( 1 - 3.36iT - 13T^{2} \) |
| 17 | \( 1 - 0.841T + 17T^{2} \) |
| 19 | \( 1 + 5.59iT - 19T^{2} \) |
| 23 | \( 1 - 2.35iT - 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 8.66iT - 31T^{2} \) |
| 37 | \( 1 - 5.15T + 37T^{2} \) |
| 41 | \( 1 + 5.74T + 41T^{2} \) |
| 43 | \( 1 - 3.32T + 43T^{2} \) |
| 47 | \( 1 - 6.43T + 47T^{2} \) |
| 53 | \( 1 + 9.64iT - 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 1.82T + 67T^{2} \) |
| 71 | \( 1 + 3.74iT - 71T^{2} \) |
| 73 | \( 1 + 0.979iT - 73T^{2} \) |
| 79 | \( 1 - 6.58T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 2.16T + 89T^{2} \) |
| 97 | \( 1 + 12.0iT - 97T^{2} \) |
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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
−8.073460843614211110209969161208, −7.15178404431294348486304423543, −6.71545566209461908695468527971, −5.71901111319406856901916113706, −5.09599284155148076154453482430, −4.37293564763306036643000757861, −3.56924393324752444742862426586, −2.64609737582266541962708031276, −1.65707217025579837944014125959, −0.68502018764535002722487633868,
0.975310658612178749888745481566, 1.98809594525683452840444207147, 2.69341646447519719899078001009, 3.91361415793534723034816462700, 4.43869649028578794834834194168, 5.38151187040492690034184224248, 5.79356314008347828904911365224, 6.77782844323124021256455555502, 7.71838477199652421798233616390, 7.87397648942102122729585178557