| L(s) = 1 | − 2·2-s − 3.89·3-s + 4·4-s + 7.79·6-s − 8·8-s + 6.20·9-s − 21.6·11-s − 15.5·12-s + 16·16-s + 28.3·17-s − 12.4·18-s + 31.6·19-s + 43.3·22-s + 31.1·24-s + 10.9·27-s − 32·32-s + 84.5·33-s − 56.7·34-s + 24.8·36-s − 63.3·38-s + 81.7·41-s + 14·43-s − 86.7·44-s − 62.3·48-s + 49·49-s − 110.·51-s − 21.8·54-s + ⋯ |
| L(s) = 1 | − 2-s − 1.29·3-s + 4-s + 1.29·6-s − 8-s + 0.689·9-s − 1.97·11-s − 1.29·12-s + 16-s + 1.67·17-s − 0.689·18-s + 1.66·19-s + 1.97·22-s + 1.29·24-s + 0.404·27-s − 32-s + 2.56·33-s − 1.67·34-s + 0.689·36-s − 1.66·38-s + 1.99·41-s + 0.325·43-s − 1.97·44-s − 1.29·48-s + 0.999·49-s − 2.17·51-s − 0.404·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5233040264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5233040264\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 3.89T + 9T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 + 21.6T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 28.3T + 289T^{2} \) |
| 19 | \( 1 - 31.6T + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 81.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 14T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 + 82T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 71.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 41.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 34.9T + 6.88e3T^{2} \) |
| 89 | \( 1 - 15.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + 94T + 9.40e3T^{2} \) |
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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
−12.00887304342222067270847484915, −11.04782728594932765000367909273, −10.37931986148220533331691829645, −9.554508018014441704341197550123, −7.963742464097824034712416175962, −7.35436727818536139680034500717, −5.82737591090727636570286472307, −5.29406282903518326366977717440, −2.90878777827577758523061980899, −0.789102809751559382473556484891,
0.789102809751559382473556484891, 2.90878777827577758523061980899, 5.29406282903518326366977717440, 5.82737591090727636570286472307, 7.35436727818536139680034500717, 7.963742464097824034712416175962, 9.554508018014441704341197550123, 10.37931986148220533331691829645, 11.04782728594932765000367909273, 12.00887304342222067270847484915
