L(s) = 1 | − 3·3-s − 2·7-s + 9·9-s − 70·11-s − 54·13-s + 22·17-s − 24·19-s + 6·21-s − 100·23-s − 27·27-s + 216·29-s − 208·31-s + 210·33-s + 254·37-s + 162·39-s − 206·41-s + 292·43-s − 320·47-s − 339·49-s − 66·51-s + 402·53-s + 72·57-s + 370·59-s − 550·61-s − 18·63-s + 728·67-s + 300·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.107·7-s + 1/3·9-s − 1.91·11-s − 1.15·13-s + 0.313·17-s − 0.289·19-s + 0.0623·21-s − 0.906·23-s − 0.192·27-s + 1.38·29-s − 1.20·31-s + 1.10·33-s + 1.12·37-s + 0.665·39-s − 0.784·41-s + 1.03·43-s − 0.993·47-s − 0.988·49-s − 0.181·51-s + 1.04·53-s + 0.167·57-s + 0.816·59-s − 1.15·61-s − 0.0359·63-s + 1.32·67-s + 0.523·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7577443361\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7577443361\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 70 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 22 T + p^{3} T^{2} \) |
| 19 | \( 1 + 24 T + p^{3} T^{2} \) |
| 23 | \( 1 + 100 T + p^{3} T^{2} \) |
| 29 | \( 1 - 216 T + p^{3} T^{2} \) |
| 31 | \( 1 + 208 T + p^{3} T^{2} \) |
| 37 | \( 1 - 254 T + p^{3} T^{2} \) |
| 41 | \( 1 + 206 T + p^{3} T^{2} \) |
| 43 | \( 1 - 292 T + p^{3} T^{2} \) |
| 47 | \( 1 + 320 T + p^{3} T^{2} \) |
| 53 | \( 1 - 402 T + p^{3} T^{2} \) |
| 59 | \( 1 - 370 T + p^{3} T^{2} \) |
| 61 | \( 1 + 550 T + p^{3} T^{2} \) |
| 67 | \( 1 - 728 T + p^{3} T^{2} \) |
| 71 | \( 1 - 540 T + p^{3} T^{2} \) |
| 73 | \( 1 + 604 T + p^{3} T^{2} \) |
| 79 | \( 1 + 792 T + p^{3} T^{2} \) |
| 83 | \( 1 - 404 T + p^{3} T^{2} \) |
| 89 | \( 1 + 938 T + p^{3} T^{2} \) |
| 97 | \( 1 + 56 T + p^{3} T^{2} \) |
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
−9.693458728127956255151787252341, −8.360960469320114074023853196207, −7.72641374550593698295202583392, −6.94481541664549009755977916950, −5.86041282461895173777167325779, −5.17651087418967106764033408254, −4.40563105664808928337795122028, −3.00024550672871204800724702444, −2.10048576885252284601413899357, −0.43415509775667737184029057643,
0.43415509775667737184029057643, 2.10048576885252284601413899357, 3.00024550672871204800724702444, 4.40563105664808928337795122028, 5.17651087418967106764033408254, 5.86041282461895173777167325779, 6.94481541664549009755977916950, 7.72641374550593698295202583392, 8.360960469320114074023853196207, 9.693458728127956255151787252341