| L(s) = 1 | − 8.77·3-s + 17.3·5-s − 26.0·7-s + 49.9·9-s + 4.22·11-s − 64.0·13-s − 151.·15-s − 48.5·17-s − 19·19-s + 228.·21-s + 92.0·23-s + 174.·25-s − 201.·27-s + 88.2·29-s − 81.9·31-s − 37.0·33-s − 451.·35-s + 23.6·37-s + 561.·39-s + 17.7·41-s − 368.·43-s + 864.·45-s − 497.·47-s + 337.·49-s + 425.·51-s + 536.·53-s + 73.2·55-s + ⋯ |
| L(s) = 1 | − 1.68·3-s + 1.54·5-s − 1.40·7-s + 1.84·9-s + 0.115·11-s − 1.36·13-s − 2.61·15-s − 0.692·17-s − 0.229·19-s + 2.37·21-s + 0.834·23-s + 1.39·25-s − 1.43·27-s + 0.564·29-s − 0.474·31-s − 0.195·33-s − 2.18·35-s + 0.104·37-s + 2.30·39-s + 0.0674·41-s − 1.30·43-s + 2.86·45-s − 1.54·47-s + 0.984·49-s + 1.16·51-s + 1.39·53-s + 0.179·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7682767863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7682767863\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 + 8.77T + 27T^{2} \) |
| 5 | \( 1 - 17.3T + 125T^{2} \) |
| 7 | \( 1 + 26.0T + 343T^{2} \) |
| 11 | \( 1 - 4.22T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 48.5T + 4.91e3T^{2} \) |
| 23 | \( 1 - 92.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 88.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 81.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 23.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 17.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 368.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 497.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 536.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 36.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 630.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 282.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 595.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 597.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 427.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 493.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 921.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.08e3T + 9.12e5T^{2} \) |
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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
−9.775358651900437767722628932714, −8.909466250171584697813728561853, −7.14621326168448255069470367164, −6.61862594344849791201250593154, −6.07300625482031912790381916961, −5.28687096676857247266586368394, −4.60335072780410334854389832680, −2.99673260301974631476974528511, −1.84338025992264241344640002805, −0.47450578254244322611080032702,
0.47450578254244322611080032702, 1.84338025992264241344640002805, 2.99673260301974631476974528511, 4.60335072780410334854389832680, 5.28687096676857247266586368394, 6.07300625482031912790381916961, 6.61862594344849791201250593154, 7.14621326168448255069470367164, 8.909466250171584697813728561853, 9.775358651900437767722628932714
