| L(s) = 1 | − 2-s + 4-s + 2·5-s − 4·7-s − 8-s − 2·10-s − 4·11-s + 4·14-s + 16-s − 2·17-s + 8·19-s + 2·20-s + 4·22-s − 25-s − 4·28-s − 6·29-s + 4·31-s − 32-s + 2·34-s − 8·35-s + 2·37-s − 8·38-s − 2·40-s − 10·41-s + 4·43-s − 4·44-s + 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 1.51·7-s − 0.353·8-s − 0.632·10-s − 1.20·11-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 1.83·19-s + 0.447·20-s + 0.852·22-s − 1/5·25-s − 0.755·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.342·34-s − 1.35·35-s + 0.328·37-s − 1.29·38-s − 0.316·40-s − 1.56·41-s + 0.609·43-s − 0.603·44-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.019266046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019266046\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.998443831037485705362471997542, −7.926120891386737881417767062013, −7.25157338538997994422876175223, −6.52936225966612524334758029220, −5.73621792555503205353069594093, −5.21952291929979392028608741127, −3.69412307098602015525916389321, −2.86490514328812342451013646175, −2.12353885842057093006820550564, −0.65392673463149137136074794790,
0.65392673463149137136074794790, 2.12353885842057093006820550564, 2.86490514328812342451013646175, 3.69412307098602015525916389321, 5.21952291929979392028608741127, 5.73621792555503205353069594093, 6.52936225966612524334758029220, 7.25157338538997994422876175223, 7.926120891386737881417767062013, 8.998443831037485705362471997542
