| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 4·11-s + 12-s + 2·13-s + 14-s + 16-s − 2·17-s − 18-s − 4·19-s − 21-s − 4·22-s + 8·23-s − 24-s − 2·26-s + 27-s − 28-s + 6·29-s − 8·31-s − 32-s + 4·33-s + 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.218·21-s − 0.852·22-s + 1.66·23-s − 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 0.696·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.500296459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.500296459\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−19.80947938140135, −19.42840723435854, −18.85746559851555, −18.06150960565033, −17.25559194004923, −16.75015998219883, −15.91604611383246, −15.24567627615569, −14.55105434570754, −13.83562083136504, −12.86475936523840, −12.33763911098049, −11.13851704739511, −10.77928151464729, −9.625809353569461, −8.993802635293948, −8.601351251693855, −7.417500049713973, −6.738926441538911, −5.952513639975337, −4.457760043065569, −3.512936926347472, −2.392858002556172, −1.086561495877088,
1.086561495877088, 2.392858002556172, 3.512936926347472, 4.457760043065569, 5.952513639975337, 6.738926441538911, 7.417500049713973, 8.601351251693855, 8.993802635293948, 9.625809353569461, 10.77928151464729, 11.13851704739511, 12.33763911098049, 12.86475936523840, 13.83562083136504, 14.55105434570754, 15.24567627615569, 15.91604611383246, 16.75015998219883, 17.25559194004923, 18.06150960565033, 18.85746559851555, 19.42840723435854, 19.80947938140135
