L(s) = 1 | + i·2-s − 0.872i·3-s − 4-s + (−2.23 + 0.146i)5-s + 0.872·6-s − 2.79i·7-s − i·8-s + 2.23·9-s + (−0.146 − 2.23i)10-s − 6.53·11-s + 0.872i·12-s − 3.33i·13-s + 2.79·14-s + (0.127 + 1.94i)15-s + 16-s − 6.39i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.503i·3-s − 0.5·4-s + (−0.997 + 0.0654i)5-s + 0.356·6-s − 1.05i·7-s − 0.353i·8-s + 0.746·9-s + (−0.0463 − 0.705i)10-s − 1.97·11-s + 0.251i·12-s − 0.924i·13-s + 0.747·14-s + (0.0329 + 0.502i)15-s + 0.250·16-s − 1.55i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0654 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0654 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.517813 - 0.484946i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.517813 - 0.484946i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (2.23 - 0.146i)T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 + 0.872iT - 3T^{2} \) |
| 7 | \( 1 + 2.79iT - 7T^{2} \) |
| 11 | \( 1 + 6.53T + 11T^{2} \) |
| 13 | \( 1 + 3.33iT - 13T^{2} \) |
| 17 | \( 1 + 6.39iT - 17T^{2} \) |
| 19 | \( 1 - 1.45T + 19T^{2} \) |
| 23 | \( 1 - 2.50iT - 23T^{2} \) |
| 29 | \( 1 + 5.98T + 29T^{2} \) |
| 37 | \( 1 + 2.11iT - 37T^{2} \) |
| 41 | \( 1 - 3.81T + 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 - 6.12iT - 47T^{2} \) |
| 53 | \( 1 + 9.07iT - 53T^{2} \) |
| 59 | \( 1 + 1.59T + 59T^{2} \) |
| 61 | \( 1 - 3.23T + 61T^{2} \) |
| 67 | \( 1 + 7.87iT - 67T^{2} \) |
| 71 | \( 1 + 0.841T + 71T^{2} \) |
| 73 | \( 1 - 12.7iT - 73T^{2} \) |
| 79 | \( 1 + 8.24T + 79T^{2} \) |
| 83 | \( 1 + 5.02iT - 83T^{2} \) |
| 89 | \( 1 - 5.93T + 89T^{2} \) |
| 97 | \( 1 + 13.5iT - 97T^{2} \) |
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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
−11.40478382340095956165531686346, −10.52176962277694681698687104229, −9.617807068805448495424959408332, −7.968473503935376662538051290873, −7.60848675835332127431508659932, −7.05857260531059180519714547056, −5.42634317922459388840786868098, −4.46356541949105314659793058310, −3.08520180544911936037125102256, −0.51272800094664941691730863848,
2.19715455140524794931988549567, 3.60710414374613342449584885031, 4.62158371325469031723019926712, 5.63711931128745475353735256669, 7.34067834334407168599691427727, 8.324690080617018533295031348094, 9.126362169065908651659885650690, 10.33145623615115330832978678768, 10.84678773446683959711578697474, 11.93849683281291160068439995005