| L(s) = 1 | − 0.644·2-s + 4.18·3-s − 7.58·4-s − 2.69·6-s + 10.0·8-s − 9.47·9-s − 47.7·11-s − 31.7·12-s − 57.2·13-s + 54.1·16-s + 36.9·17-s + 6.10·18-s + 30.7·19-s + 30.7·22-s − 53.1·23-s + 42.0·24-s + 36.8·26-s − 152.·27-s − 195.·29-s + 257.·31-s − 115.·32-s − 199.·33-s − 23.8·34-s + 71.8·36-s − 346.·37-s − 19.8·38-s − 239.·39-s + ⋯ |
| L(s) = 1 | − 0.227·2-s + 0.805·3-s − 0.948·4-s − 0.183·6-s + 0.443·8-s − 0.350·9-s − 1.30·11-s − 0.763·12-s − 1.22·13-s + 0.846·16-s + 0.527·17-s + 0.0799·18-s + 0.371·19-s + 0.298·22-s − 0.481·23-s + 0.357·24-s + 0.278·26-s − 1.08·27-s − 1.25·29-s + 1.49·31-s − 0.637·32-s − 1.05·33-s − 0.120·34-s + 0.332·36-s − 1.53·37-s − 0.0846·38-s − 0.983·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.104947859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.104947859\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.644T + 8T^{2} \) |
| 3 | \( 1 - 4.18T + 27T^{2} \) |
| 11 | \( 1 + 47.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 57.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 36.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 30.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 53.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 195.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 257.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 346.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 267.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 176.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 311.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 492.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 98.7T + 2.05e5T^{2} \) |
| 61 | \( 1 - 82.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 654.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 779.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 829.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 769.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 613.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 457.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.41e3T + 9.12e5T^{2} \) |
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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
−9.300851006824140403385953907033, −8.544736713892999301158853718966, −7.81142350352958246491036833155, −7.36294886727581643074684553247, −5.70131974715951036443846473144, −5.14647697201649844817229424259, −4.08653988054015793670117507008, −3.04980353578761495428201239968, −2.18942327824191924712059272771, −0.51381402720665545207744093757,
0.51381402720665545207744093757, 2.18942327824191924712059272771, 3.04980353578761495428201239968, 4.08653988054015793670117507008, 5.14647697201649844817229424259, 5.70131974715951036443846473144, 7.36294886727581643074684553247, 7.81142350352958246491036833155, 8.544736713892999301158853718966, 9.300851006824140403385953907033
