| L(s) = 1 | + 3-s − 7-s + 9-s − 11-s + 13-s − 2·17-s − 5·19-s − 21-s − 2·23-s + 27-s − 6·29-s + 3·31-s − 33-s − 2·37-s + 39-s − 6·41-s + 3·43-s − 8·47-s − 6·49-s − 2·51-s + 6·53-s − 5·57-s + 8·59-s − 11·61-s − 63-s − 3·67-s − 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 0.485·17-s − 1.14·19-s − 0.218·21-s − 0.417·23-s + 0.192·27-s − 1.11·29-s + 0.538·31-s − 0.174·33-s − 0.328·37-s + 0.160·39-s − 0.937·41-s + 0.457·43-s − 1.16·47-s − 6/7·49-s − 0.280·51-s + 0.824·53-s − 0.662·57-s + 1.04·59-s − 1.40·61-s − 0.125·63-s − 0.366·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−8.340734512559084445958699000897, −7.62118827273166531035747567233, −6.74134220416492974461453071086, −6.15856850855926615218537670663, −5.14961933697282774915651653425, −4.24154292773293065296132660476, −3.50726520424855967033275458176, −2.55157422395953898343166693422, −1.66741251986951095564782427061, 0,
1.66741251986951095564782427061, 2.55157422395953898343166693422, 3.50726520424855967033275458176, 4.24154292773293065296132660476, 5.14961933697282774915651653425, 6.15856850855926615218537670663, 6.74134220416492974461453071086, 7.62118827273166531035747567233, 8.340734512559084445958699000897
