| L(s) = 1 | − 2-s + 4-s + 4.41·5-s − 8-s − 4.41·10-s − 11-s + 4.24·13-s + 16-s − 2.82·17-s − 5.82·19-s + 4.41·20-s + 22-s + 5.24·23-s + 14.4·25-s − 4.24·26-s − 0.242·29-s − 7.24·31-s − 32-s + 2.82·34-s + 5.24·37-s + 5.82·38-s − 4.41·40-s + 6.17·41-s − 6.48·43-s − 44-s − 5.24·46-s + 11.6·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.97·5-s − 0.353·8-s − 1.39·10-s − 0.301·11-s + 1.17·13-s + 0.250·16-s − 0.685·17-s − 1.33·19-s + 0.987·20-s + 0.213·22-s + 1.09·23-s + 2.89·25-s − 0.832·26-s − 0.0450·29-s − 1.30·31-s − 0.176·32-s + 0.485·34-s + 0.861·37-s + 0.945·38-s − 0.697·40-s + 0.963·41-s − 0.988·43-s − 0.150·44-s − 0.772·46-s + 1.70·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.008334929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.008334929\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4.41T + 5T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 5.82T + 19T^{2} \) |
| 23 | \( 1 - 5.24T + 23T^{2} \) |
| 29 | \( 1 + 0.242T + 29T^{2} \) |
| 31 | \( 1 + 7.24T + 31T^{2} \) |
| 37 | \( 1 - 5.24T + 37T^{2} \) |
| 41 | \( 1 - 6.17T + 41T^{2} \) |
| 43 | \( 1 + 6.48T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 14.8T + 61T^{2} \) |
| 67 | \( 1 + 1.75T + 67T^{2} \) |
| 71 | \( 1 + 1.24T + 71T^{2} \) |
| 73 | \( 1 + 3.17T + 73T^{2} \) |
| 79 | \( 1 + 6.48T + 79T^{2} \) |
| 83 | \( 1 + 3.89T + 83T^{2} \) |
| 89 | \( 1 - 3.34T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−8.782903023433583479286118555916, −8.587765973830842118279040398234, −7.17563632660720277061192917403, −6.57060781095275917307227366040, −5.87006945874481112881596803230, −5.29995501718085783151563751578, −4.04075434980751448690546167187, −2.66429902884248853369298034767, −2.05707166495018111764908864067, −1.04125364241775240333172496616,
1.04125364241775240333172496616, 2.05707166495018111764908864067, 2.66429902884248853369298034767, 4.04075434980751448690546167187, 5.29995501718085783151563751578, 5.87006945874481112881596803230, 6.57060781095275917307227366040, 7.17563632660720277061192917403, 8.587765973830842118279040398234, 8.782903023433583479286118555916
