L(s) = 1 | + 4-s + i·7-s + 16-s + i·28-s + 2i·37-s − 2i·43-s − 49-s + 64-s − 2i·67-s − 2·79-s + 2·109-s + i·112-s + ⋯ |
L(s) = 1 | + 4-s + i·7-s + 16-s + i·28-s + 2i·37-s − 2i·43-s − 49-s + 64-s − 2i·67-s − 2·79-s + 2·109-s + i·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.405897070\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.405897070\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 2 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 2iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 2iT - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 2iT - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + 2T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.751939668581427896467730196113, −8.785094445161926091883884528579, −8.120982520058901001726358376490, −7.20254116341639988099969785528, −6.42567958369940700772505248296, −5.72494990932537743023290213341, −4.85669050903742721122450969610, −3.47715774174445045269970651000, −2.61732237927340724392963698213, −1.66337124634595529707566199574,
1.27033692199437885483573765158, 2.49888726773389951467309249421, 3.53510049815431303241984854106, 4.44484030762349950054090385695, 5.62711434928389653058091435628, 6.41633951449973542666162314434, 7.25543411897011803892344680586, 7.69533806277893102285417128286, 8.696821163695744809947302107927, 9.777662703165187287566671656296