| L(s) = 1 | − 0.811·3-s + 5-s + 7-s − 2.34·9-s + 5.46·11-s − 13-s − 0.811·15-s − 0.311·17-s − 5.28·19-s − 0.811·21-s + 3.74·23-s + 25-s + 4.33·27-s − 4.34·29-s − 4.65·31-s − 4.43·33-s + 35-s + 1.88·37-s + 0.811·39-s + 10.8·41-s − 9.39·43-s − 2.34·45-s − 1.08·47-s + 49-s + 0.252·51-s + 9.41·53-s + 5.46·55-s + ⋯ |
| L(s) = 1 | − 0.468·3-s + 0.447·5-s + 0.377·7-s − 0.780·9-s + 1.64·11-s − 0.277·13-s − 0.209·15-s − 0.0754·17-s − 1.21·19-s − 0.177·21-s + 0.781·23-s + 0.200·25-s + 0.834·27-s − 0.806·29-s − 0.836·31-s − 0.772·33-s + 0.169·35-s + 0.309·37-s + 0.129·39-s + 1.68·41-s − 1.43·43-s − 0.348·45-s − 0.158·47-s + 0.142·49-s + 0.0353·51-s + 1.29·53-s + 0.737·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.851477322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.851477322\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 0.811T + 3T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 17 | \( 1 + 0.311T + 17T^{2} \) |
| 19 | \( 1 + 5.28T + 19T^{2} \) |
| 23 | \( 1 - 3.74T + 23T^{2} \) |
| 29 | \( 1 + 4.34T + 29T^{2} \) |
| 31 | \( 1 + 4.65T + 31T^{2} \) |
| 37 | \( 1 - 1.88T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 9.39T + 43T^{2} \) |
| 47 | \( 1 + 1.08T + 47T^{2} \) |
| 53 | \( 1 - 9.41T + 53T^{2} \) |
| 59 | \( 1 - 8.65T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 7.06T + 67T^{2} \) |
| 71 | \( 1 + 1.98T + 71T^{2} \) |
| 73 | \( 1 + 3.77T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 4.77T + 83T^{2} \) |
| 89 | \( 1 + 2.62T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−8.006882760566142929468789527300, −6.85685056125593278847678283713, −6.66676919403889743221638000326, −5.72982981217637088533052589653, −5.29019548045985885469729718754, −4.31036272529109817880655483886, −3.70880592551662188707901959601, −2.57871955205970795624783265926, −1.76998161579980624598639532714, −0.71429569605819886136834791734,
0.71429569605819886136834791734, 1.76998161579980624598639532714, 2.57871955205970795624783265926, 3.70880592551662188707901959601, 4.31036272529109817880655483886, 5.29019548045985885469729718754, 5.72982981217637088533052589653, 6.66676919403889743221638000326, 6.85685056125593278847678283713, 8.006882760566142929468789527300
