L(s) = 1 | − 2.32i·2-s + 1.14i·3-s − 3.40·4-s + (−2.01 − 0.975i)5-s + 2.67·6-s + 0.143i·7-s + 3.26i·8-s + 1.68·9-s + (−2.26 + 4.67i)10-s − 2.81·11-s − 3.90i·12-s + 1.70i·13-s + 0.333·14-s + (1.12 − 2.31i)15-s + 0.778·16-s − 3.55i·17-s + ⋯ |
L(s) = 1 | − 1.64i·2-s + 0.663i·3-s − 1.70·4-s + (−0.899 − 0.436i)5-s + 1.09·6-s + 0.0541i·7-s + 1.15i·8-s + 0.560·9-s + (−0.717 + 1.47i)10-s − 0.848·11-s − 1.12i·12-s + 0.473i·13-s + 0.0890·14-s + (0.289 − 0.596i)15-s + 0.194·16-s − 0.863i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9346673089\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9346673089\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.01 + 0.975i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 2.32iT - 2T^{2} \) |
| 3 | \( 1 - 1.14iT - 3T^{2} \) |
| 7 | \( 1 - 0.143iT - 7T^{2} \) |
| 11 | \( 1 + 2.81T + 11T^{2} \) |
| 13 | \( 1 - 1.70iT - 13T^{2} \) |
| 17 | \( 1 + 3.55iT - 17T^{2} \) |
| 23 | \( 1 - 7.19iT - 23T^{2} \) |
| 29 | \( 1 + 7.57T + 29T^{2} \) |
| 31 | \( 1 - 4.84T + 31T^{2} \) |
| 37 | \( 1 + 9.49iT - 37T^{2} \) |
| 41 | \( 1 + 0.187T + 41T^{2} \) |
| 43 | \( 1 - 10.9iT - 43T^{2} \) |
| 47 | \( 1 - 3.67iT - 47T^{2} \) |
| 53 | \( 1 - 1.64iT - 53T^{2} \) |
| 59 | \( 1 - 5.36T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 6.00iT - 67T^{2} \) |
| 71 | \( 1 - 0.540T + 71T^{2} \) |
| 73 | \( 1 - 10.4iT - 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 7.59iT - 83T^{2} \) |
| 89 | \( 1 + 1.44T + 89T^{2} \) |
| 97 | \( 1 + 8.80iT - 97T^{2} \) |
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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
−9.508802188292352508055540726137, −8.880186178843874761516159958843, −7.78590935637536342854211536308, −7.13208466763549047804468213780, −5.44713703629328093471189429487, −4.73179617280780989106962661478, −3.99106529126715871651452671919, −3.37885304981074044585133096056, −2.26243529655563858403187829516, −0.999351101644443375498187360074,
0.43778532456184738499047861695, 2.34990089794977392307894203830, 3.76361898826622716538818364021, 4.60707618135933993724996681763, 5.51784847484869940907618851850, 6.47577216174902565933356958786, 6.95221287183461382317241648746, 7.68192099631096698859619176383, 8.175404500574204322161636557162, 8.732275059658487700372277720090