| L(s) = 1 | + 2.82·3-s + 1.41·5-s − 4·7-s + 5.00·9-s + 2.82·11-s + 4.24·13-s + 4.00·15-s + 2.82·19-s − 11.3·21-s − 4·23-s − 2.99·25-s + 5.65·27-s + 4.24·29-s + 8·31-s + 8.00·33-s − 5.65·35-s − 1.41·37-s + 12·39-s + 8·41-s − 2.82·43-s + 7.07·45-s − 8·47-s + 9·49-s + 1.41·53-s + 4.00·55-s + 8.00·57-s − 8.48·59-s + ⋯ |
| L(s) = 1 | + 1.63·3-s + 0.632·5-s − 1.51·7-s + 1.66·9-s + 0.852·11-s + 1.17·13-s + 1.03·15-s + 0.648·19-s − 2.46·21-s − 0.834·23-s − 0.599·25-s + 1.08·27-s + 0.787·29-s + 1.43·31-s + 1.39·33-s − 0.956·35-s − 0.232·37-s + 1.92·39-s + 1.24·41-s − 0.431·43-s + 1.05·45-s − 1.16·47-s + 1.28·49-s + 0.194·53-s + 0.539·55-s + 1.05·57-s − 1.10·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.929658053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.929658053\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 1.41T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 + 2.82T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 16T + 97T^{2} \) |
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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
−9.777200312557125456398200364677, −9.141942118401096359783638849359, −8.505502495890198784157155837380, −7.58037971453554779535647467079, −6.48063372378443633260847244741, −6.03081826493118168330459260519, −4.25651787133443039171123656142, −3.43063009781623034885707427323, −2.75527980827503035579234444624, −1.46167098212996603597209037968,
1.46167098212996603597209037968, 2.75527980827503035579234444624, 3.43063009781623034885707427323, 4.25651787133443039171123656142, 6.03081826493118168330459260519, 6.48063372378443633260847244741, 7.58037971453554779535647467079, 8.505502495890198784157155837380, 9.141942118401096359783638849359, 9.777200312557125456398200364677
