| L(s) = 1 | + 3·3-s − 16·5-s − 28·7-s + 9·9-s − 34·11-s − 13·13-s − 48·15-s + 138·17-s − 108·19-s − 84·21-s + 52·23-s + 131·25-s + 27·27-s − 190·29-s + 176·31-s − 102·33-s + 448·35-s + 342·37-s − 39·39-s + 240·41-s + 140·43-s − 144·45-s − 454·47-s + 441·49-s + 414·51-s + 198·53-s + 544·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.43·5-s − 1.51·7-s + 1/3·9-s − 0.931·11-s − 0.277·13-s − 0.826·15-s + 1.96·17-s − 1.30·19-s − 0.872·21-s + 0.471·23-s + 1.04·25-s + 0.192·27-s − 1.21·29-s + 1.01·31-s − 0.538·33-s + 2.16·35-s + 1.51·37-s − 0.160·39-s + 0.914·41-s + 0.496·43-s − 0.477·45-s − 1.40·47-s + 9/7·49-s + 1.13·51-s + 0.513·53-s + 1.33·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.042113349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.042113349\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 13 | \( 1 + p T \) |
| good | 5 | \( 1 + 16 T + p^{3} T^{2} \) |
| 7 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 138 T + p^{3} T^{2} \) |
| 19 | \( 1 + 108 T + p^{3} T^{2} \) |
| 23 | \( 1 - 52 T + p^{3} T^{2} \) |
| 29 | \( 1 + 190 T + p^{3} T^{2} \) |
| 31 | \( 1 - 176 T + p^{3} T^{2} \) |
| 37 | \( 1 - 342 T + p^{3} T^{2} \) |
| 41 | \( 1 - 240 T + p^{3} T^{2} \) |
| 43 | \( 1 - 140 T + p^{3} T^{2} \) |
| 47 | \( 1 + 454 T + p^{3} T^{2} \) |
| 53 | \( 1 - 198 T + p^{3} T^{2} \) |
| 59 | \( 1 - 154 T + p^{3} T^{2} \) |
| 61 | \( 1 - 34 T + p^{3} T^{2} \) |
| 67 | \( 1 - 656 T + p^{3} T^{2} \) |
| 71 | \( 1 + 550 T + p^{3} T^{2} \) |
| 73 | \( 1 - 614 T + p^{3} T^{2} \) |
| 79 | \( 1 + 8 T + p^{3} T^{2} \) |
| 83 | \( 1 + 762 T + p^{3} T^{2} \) |
| 89 | \( 1 + 444 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1022 T + p^{3} T^{2} \) |
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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
−10.06732199496079452808008205346, −9.432183438316352241773817880962, −8.209824998995796663304952843449, −7.73985299178608377464471858039, −6.84702345346502519248766304689, −5.69882618263101134775224393075, −4.30990718041491424216125419138, −3.43956401964249081470876167175, −2.70298490649029355636659182918, −0.56079968043594391293564547524,
0.56079968043594391293564547524, 2.70298490649029355636659182918, 3.43956401964249081470876167175, 4.30990718041491424216125419138, 5.69882618263101134775224393075, 6.84702345346502519248766304689, 7.73985299178608377464471858039, 8.209824998995796663304952843449, 9.432183438316352241773817880962, 10.06732199496079452808008205346
