L(s) = 1 | − 3-s − 4-s + 9-s + 12-s + 13-s + 16-s − 17-s + 2·23-s + 25-s − 27-s + 2·29-s − 36-s − 39-s − 2·43-s − 48-s − 49-s + 51-s − 52-s − 64-s + 68-s − 2·69-s − 75-s + 81-s − 2·87-s − 2·92-s − 100-s + 2·103-s + ⋯ |
L(s) = 1 | − 3-s − 4-s + 9-s + 12-s + 13-s + 16-s − 17-s + 2·23-s + 25-s − 27-s + 2·29-s − 36-s − 39-s − 2·43-s − 48-s − 49-s + 51-s − 52-s − 64-s + 68-s − 2·69-s − 75-s + 81-s − 2·87-s − 2·92-s − 100-s + 2·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5870665681\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5870665681\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )^{2} \) |
| 29 | \( ( 1 - T )^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 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
−10.76442110915722872977724570457, −9.992514726477694523288268333026, −8.952391434061781417043692343644, −8.363591650842082987448856734975, −6.95696202508489808699934997289, −6.28380074178497794556108203766, −5.04125243326520469064759691898, −4.60021500321488830639239683246, −3.28988102135128638814816860891, −1.11519097983291239832997370225,
1.11519097983291239832997370225, 3.28988102135128638814816860891, 4.60021500321488830639239683246, 5.04125243326520469064759691898, 6.28380074178497794556108203766, 6.95696202508489808699934997289, 8.363591650842082987448856734975, 8.952391434061781417043692343644, 9.992514726477694523288268333026, 10.76442110915722872977724570457