| L(s) = 1 | − 2-s + 3.10·3-s + 4-s + 2.70·5-s − 3.10·6-s + 1.47·7-s − 8-s + 6.63·9-s − 2.70·10-s + 3.10·12-s − 0.908·13-s − 1.47·14-s + 8.40·15-s + 16-s − 3.03·17-s − 6.63·18-s + 1.45·19-s + 2.70·20-s + 4.56·21-s − 23-s − 3.10·24-s + 2.32·25-s + 0.908·26-s + 11.2·27-s + 1.47·28-s + 8.39·29-s − 8.40·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.79·3-s + 0.5·4-s + 1.21·5-s − 1.26·6-s + 0.555·7-s − 0.353·8-s + 2.21·9-s − 0.856·10-s + 0.895·12-s − 0.252·13-s − 0.392·14-s + 2.16·15-s + 0.250·16-s − 0.735·17-s − 1.56·18-s + 0.334·19-s + 0.605·20-s + 0.995·21-s − 0.208·23-s − 0.633·24-s + 0.465·25-s + 0.178·26-s + 2.16·27-s + 0.277·28-s + 1.55·29-s − 1.53·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5566 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5566 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.067244831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.067244831\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 3.10T + 3T^{2} \) |
| 5 | \( 1 - 2.70T + 5T^{2} \) |
| 7 | \( 1 - 1.47T + 7T^{2} \) |
| 13 | \( 1 + 0.908T + 13T^{2} \) |
| 17 | \( 1 + 3.03T + 17T^{2} \) |
| 19 | \( 1 - 1.45T + 19T^{2} \) |
| 29 | \( 1 - 8.39T + 29T^{2} \) |
| 31 | \( 1 - 5.01T + 31T^{2} \) |
| 37 | \( 1 - 1.79T + 37T^{2} \) |
| 41 | \( 1 - 0.0162T + 41T^{2} \) |
| 43 | \( 1 + 6.50T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 7.30T + 53T^{2} \) |
| 59 | \( 1 + 0.224T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 8.06T + 71T^{2} \) |
| 73 | \( 1 - 8.36T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 5.48T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 2.29T + 97T^{2} \) |
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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
−8.339960008789889768593016940419, −7.75456035661677671992612236927, −6.84485921695076562591355447166, −6.36165623370881285901263921677, −5.16284525042376655179157871923, −4.41033643951703963125972076304, −3.31130234786668961438455458881, −2.53364479380369180383353177810, −2.01217541899927968368698914040, −1.20940155758188146385880823992,
1.20940155758188146385880823992, 2.01217541899927968368698914040, 2.53364479380369180383353177810, 3.31130234786668961438455458881, 4.41033643951703963125972076304, 5.16284525042376655179157871923, 6.36165623370881285901263921677, 6.84485921695076562591355447166, 7.75456035661677671992612236927, 8.339960008789889768593016940419
