| L(s) = 1 | + 2i·2-s + 3i·3-s − 4·4-s − 6·6-s − 7i·7-s − 8i·8-s − 9·9-s − 12i·12-s + 26i·13-s + 14·14-s + 16·16-s − 18i·17-s − 18i·18-s − 92·19-s + 21·21-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.377i·7-s − 0.353i·8-s − 0.333·9-s − 0.288i·12-s + 0.554i·13-s + 0.267·14-s + 0.250·16-s − 0.256i·17-s − 0.235i·18-s − 1.11·19-s + 0.218·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.479242460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.479242460\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7iT \) |
| good | 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 - 26iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 18iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 92T + 6.85e3T^{2} \) |
| 23 | \( 1 - 1.21e4T^{2} \) |
| 29 | \( 1 - 6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 410iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 174T + 6.89e4T^{2} \) |
| 43 | \( 1 - 248iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 420iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 102iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 588T + 2.05e5T^{2} \) |
| 61 | \( 1 - 650T + 2.26e5T^{2} \) |
| 67 | \( 1 + 152iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 168T + 3.57e5T^{2} \) |
| 73 | \( 1 + 610iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 684iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 834T + 7.04e5T^{2} \) |
| 97 | \( 1 + 110iT - 9.12e5T^{2} \) |
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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
−9.428712119352182343826677400094, −8.813300547246414527212662215880, −7.928099136960535259483017847649, −7.03315385832818460396791929396, −6.24152549020998964330523329032, −5.28772071252497435373384869198, −4.37025237511892437856892159129, −3.66979604113024556571203763163, −2.20024820603368514896279211342, −0.48559575863204253630972390157,
0.834702101896117258357679907546, 2.02399222979672837627205515814, 2.90268356798282158583098580144, 4.02711178236988241265120013929, 5.12325342945012997279435123916, 6.04256384218273206046609311524, 6.92428786950056481267015988856, 8.100363265798700095998177848210, 8.543508375563387912046061180746, 9.556963018865082973074995203754
