| L(s) = 1 | + 3-s + 3.41·5-s + 9-s − 0.828·11-s − 4.24·13-s + 3.41·15-s + 7.41·17-s − 6.82·19-s + 4.82·23-s + 6.65·25-s + 27-s + 2.82·29-s + 2.82·31-s − 0.828·33-s − 1.65·37-s − 4.24·39-s + 10.2·41-s + 11.3·43-s + 3.41·45-s + 4.48·47-s + 7.41·51-s − 2·53-s − 2.82·55-s − 6.82·57-s − 8.48·59-s + 11.0·61-s − 14.4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.52·5-s + 0.333·9-s − 0.249·11-s − 1.17·13-s + 0.881·15-s + 1.79·17-s − 1.56·19-s + 1.00·23-s + 1.33·25-s + 0.192·27-s + 0.525·29-s + 0.508·31-s − 0.144·33-s − 0.272·37-s − 0.679·39-s + 1.59·41-s + 1.72·43-s + 0.508·45-s + 0.654·47-s + 1.03·51-s − 0.274·53-s − 0.381·55-s − 0.904·57-s − 1.10·59-s + 1.41·61-s − 1.79·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.990466645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.990466645\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.41T + 5T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 7.41T + 17T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 + 1.65T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 4.48T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 7.75T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 5.75T + 89T^{2} \) |
| 97 | \( 1 + 0.242T + 97T^{2} \) |
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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
−9.259618155289063275474409776343, −8.233023528620307269063895365542, −7.50004668780617771490234564208, −6.64030971592329048853322092672, −5.80532898145921190008445452790, −5.14670847304765914727283422717, −4.18509539851081598699484819786, −2.81306985594316566180527736166, −2.36855787342397343859960885168, −1.16894334555770476561511447763,
1.16894334555770476561511447763, 2.36855787342397343859960885168, 2.81306985594316566180527736166, 4.18509539851081598699484819786, 5.14670847304765914727283422717, 5.80532898145921190008445452790, 6.64030971592329048853322092672, 7.50004668780617771490234564208, 8.233023528620307269063895365542, 9.259618155289063275474409776343
