L(s) = 1 | + 0.445i·2-s − 1.80i·3-s + 0.801·4-s + 0.801·6-s − 1.24i·7-s + 0.801i·8-s − 2.24·9-s − 0.445·11-s − 1.44i·12-s + 0.554·14-s + 0.445·16-s − i·18-s − 2.24·21-s − 0.198i·22-s + 1.44·24-s + ⋯ |
L(s) = 1 | + 0.445i·2-s − 1.80i·3-s + 0.801·4-s + 0.801·6-s − 1.24i·7-s + 0.801i·8-s − 2.24·9-s − 0.445·11-s − 1.44i·12-s + 0.554·14-s + 0.445·16-s − i·18-s − 2.24·21-s − 0.198i·22-s + 1.44·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.174059776\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.174059776\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 43 | \( 1 - iT \) |
good | 2 | \( 1 - 0.445iT - T^{2} \) |
| 3 | \( 1 + 1.80iT - T^{2} \) |
| 7 | \( 1 + 1.24iT - T^{2} \) |
| 11 | \( 1 + 0.445T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.24T + T^{2} \) |
| 37 | \( 1 - 0.445iT - T^{2} \) |
| 41 | \( 1 + 1.80T + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - 1.80T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.80iT - T^{2} \) |
| 79 | \( 1 - 0.445T + T^{2} \) |
| 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
−10.04802328869678987004909067872, −8.485398056651046271666378763807, −7.931899453306413756839671331984, −7.28071855681999890402860631598, −6.70306150457009678744768019317, −6.09917469342986398595481584010, −4.96625506269522340318216423856, −3.28785505287123495977398712293, −2.23417876000048743471810051149, −1.13053787489434146946129715977,
2.30315987107524134514308809589, 3.06804972307304809422523897512, 3.99524390822339168955550715852, 5.15377866888966028538587975330, 5.71559322492703534722970612302, 6.76794693437643361978568397665, 8.198695376500249747867591820525, 8.834862171582189859377287698533, 9.765924483196383908707569251342, 10.22316143784553140164254018541