| L(s) = 1 | − 2-s + 4-s − 2.74·5-s + 0.881·7-s − 8-s + 2.74·10-s − 0.654·11-s + 2.06·13-s − 0.881·14-s + 16-s + 5.11·17-s − 5.28·19-s − 2.74·20-s + 0.654·22-s − 3.61·23-s + 2.52·25-s − 2.06·26-s + 0.881·28-s − 1.25·29-s + 3.23·31-s − 32-s − 5.11·34-s − 2.41·35-s + 4.59·37-s + 5.28·38-s + 2.74·40-s + 0.963·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.22·5-s + 0.333·7-s − 0.353·8-s + 0.867·10-s − 0.197·11-s + 0.573·13-s − 0.235·14-s + 0.250·16-s + 1.24·17-s − 1.21·19-s − 0.613·20-s + 0.139·22-s − 0.753·23-s + 0.505·25-s − 0.405·26-s + 0.166·28-s − 0.233·29-s + 0.581·31-s − 0.176·32-s − 0.877·34-s − 0.408·35-s + 0.754·37-s + 0.856·38-s + 0.433·40-s + 0.150·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 + 2.74T + 5T^{2} \) |
| 7 | \( 1 - 0.881T + 7T^{2} \) |
| 11 | \( 1 + 0.654T + 11T^{2} \) |
| 13 | \( 1 - 2.06T + 13T^{2} \) |
| 17 | \( 1 - 5.11T + 17T^{2} \) |
| 19 | \( 1 + 5.28T + 19T^{2} \) |
| 23 | \( 1 + 3.61T + 23T^{2} \) |
| 29 | \( 1 + 1.25T + 29T^{2} \) |
| 31 | \( 1 - 3.23T + 31T^{2} \) |
| 37 | \( 1 - 4.59T + 37T^{2} \) |
| 41 | \( 1 - 0.963T + 41T^{2} \) |
| 43 | \( 1 - 5.50T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 6.64T + 53T^{2} \) |
| 59 | \( 1 - 9.13T + 59T^{2} \) |
| 61 | \( 1 - 7.23T + 61T^{2} \) |
| 67 | \( 1 - 2.54T + 67T^{2} \) |
| 71 | \( 1 - 8.61T + 71T^{2} \) |
| 73 | \( 1 - 5.32T + 73T^{2} \) |
| 79 | \( 1 - 1.75T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 1.52T + 97T^{2} \) |
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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
−7.78679707633113118104512655679, −6.94763574286940926502760215462, −6.26161269341230211835425574218, −5.48817885125425184704557149260, −4.50888331301112532710708586086, −3.85528407353236197545156960852, −3.13556561740941262829677438538, −2.09592095627196113129488150130, −1.05988459654285081280349101946, 0,
1.05988459654285081280349101946, 2.09592095627196113129488150130, 3.13556561740941262829677438538, 3.85528407353236197545156960852, 4.50888331301112532710708586086, 5.48817885125425184704557149260, 6.26161269341230211835425574218, 6.94763574286940926502760215462, 7.78679707633113118104512655679
