| L(s) = 1 | − 2·4-s − 8·7-s + 36·13-s + 4·16-s + 48·19-s + 16·28-s + 8·31-s + 112·37-s + 160·43-s − 50·49-s − 72·52-s + 220·61-s − 8·64-s − 64·67-s − 92·73-s − 96·76-s − 72·79-s − 288·91-s − 28·97-s − 136·103-s + 272·109-s − 32·112-s + 114·121-s − 16·124-s + 127-s + 131-s − 384·133-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 8/7·7-s + 2.76·13-s + 1/4·16-s + 2.52·19-s + 4/7·28-s + 8/31·31-s + 3.02·37-s + 3.72·43-s − 1.02·49-s − 1.38·52-s + 3.60·61-s − 1/8·64-s − 0.955·67-s − 1.26·73-s − 1.26·76-s − 0.911·79-s − 3.16·91-s − 0.288·97-s − 1.32·103-s + 2.49·109-s − 2/7·112-s + 0.942·121-s − 0.129·124-s + 0.00787·127-s + 0.00763·131-s − 2.88·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.616793322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.616793322\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 114 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 576 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 510 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 224 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2784 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 80 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3618 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5600 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3090 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 7490 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 36 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 13746 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 12480 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.05911471795338420190503484606, −10.80307141943373576788978479105, −9.946040128698342078468240179949, −9.805788438053003505914599375215, −9.195078659814883246740338050096, −9.126493883799602819098283891383, −8.255237386908147828976697700201, −8.142298526002289970192637215390, −7.33040936178153298637179560579, −7.07179134660151382391339825068, −6.05248727635176297107477836096, −6.02850853799443993156627975139, −5.71606397328536232569676668185, −4.80151418707651568730771595489, −4.07158327363988806282788583782, −3.73432563504302152485440846344, −3.14187023941735518812609087728, −2.59749466311685052474553599979, −1.04969804066432428537491198664, −0.947653413572085349838215185105,
0.947653413572085349838215185105, 1.04969804066432428537491198664, 2.59749466311685052474553599979, 3.14187023941735518812609087728, 3.73432563504302152485440846344, 4.07158327363988806282788583782, 4.80151418707651568730771595489, 5.71606397328536232569676668185, 6.02850853799443993156627975139, 6.05248727635176297107477836096, 7.07179134660151382391339825068, 7.33040936178153298637179560579, 8.142298526002289970192637215390, 8.255237386908147828976697700201, 9.126493883799602819098283891383, 9.195078659814883246740338050096, 9.805788438053003505914599375215, 9.946040128698342078468240179949, 10.80307141943373576788978479105, 11.05911471795338420190503484606
