| L(s) = 1 | + 2-s + 3-s + 4-s − 2.34·5-s + 6-s + 8-s + 9-s − 2.34·10-s − 3.67·11-s + 12-s + 13-s − 2.34·15-s + 16-s + 5.67·17-s + 18-s − 2.34·19-s − 2.34·20-s − 3.67·22-s + 4.34·23-s + 24-s + 0.521·25-s + 26-s + 27-s + 9.08·29-s − 2.34·30-s − 3.32·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.05·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.743·10-s − 1.10·11-s + 0.288·12-s + 0.277·13-s − 0.606·15-s + 0.250·16-s + 1.37·17-s + 0.235·18-s − 0.539·19-s − 0.525·20-s − 0.783·22-s + 0.906·23-s + 0.204·24-s + 0.104·25-s + 0.196·26-s + 0.192·27-s + 1.68·29-s − 0.429·30-s − 0.596·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.043054708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.043054708\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2.34T + 5T^{2} \) |
| 11 | \( 1 + 3.67T + 11T^{2} \) |
| 17 | \( 1 - 5.67T + 17T^{2} \) |
| 19 | \( 1 + 2.34T + 19T^{2} \) |
| 23 | \( 1 - 4.34T + 23T^{2} \) |
| 29 | \( 1 - 9.08T + 29T^{2} \) |
| 31 | \( 1 + 3.32T + 31T^{2} \) |
| 37 | \( 1 - 3.89T + 37T^{2} \) |
| 41 | \( 1 + 1.90T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 2.91T + 47T^{2} \) |
| 53 | \( 1 + 0.737T + 53T^{2} \) |
| 59 | \( 1 + 3.32T + 59T^{2} \) |
| 61 | \( 1 + 3.80T + 61T^{2} \) |
| 67 | \( 1 - 9.28T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 + 1.65T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 7.10T + 89T^{2} \) |
| 97 | \( 1 + 6.51T + 97T^{2} \) |
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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
−8.031783922495434801998420566378, −7.972924802165420632550395695534, −7.13330402961038964418383707093, −6.26802805500544286820061278959, −5.30569446169539888386331748100, −4.64188867289458552082869869094, −3.77085333571013212487029170989, −3.14637926270568488259218886023, −2.35189950072260530061487781239, −0.891669119643008232292549722552,
0.891669119643008232292549722552, 2.35189950072260530061487781239, 3.14637926270568488259218886023, 3.77085333571013212487029170989, 4.64188867289458552082869869094, 5.30569446169539888386331748100, 6.26802805500544286820061278959, 7.13330402961038964418383707093, 7.972924802165420632550395695534, 8.031783922495434801998420566378
