| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 12.2·5-s − 6·6-s + 1.48·7-s − 8·8-s + 9·9-s + 24.4·10-s − 15.6·11-s + 12·12-s − 55.5·13-s − 2.96·14-s − 36.6·15-s + 16·16-s − 46.4·17-s − 18·18-s − 48.9·20-s + 4.44·21-s + 31.3·22-s − 115.·23-s − 24·24-s + 24.5·25-s + 111.·26-s + 27·27-s + 5.92·28-s + 14.6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.09·5-s − 0.408·6-s + 0.0799·7-s − 0.353·8-s + 0.333·9-s + 0.773·10-s − 0.429·11-s + 0.288·12-s − 1.18·13-s − 0.0565·14-s − 0.631·15-s + 0.250·16-s − 0.662·17-s − 0.235·18-s − 0.546·20-s + 0.0461·21-s + 0.303·22-s − 1.05·23-s − 0.204·24-s + 0.196·25-s + 0.837·26-s + 0.192·27-s + 0.0399·28-s + 0.0940·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6194839150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6194839150\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 12.2T + 125T^{2} \) |
| 7 | \( 1 - 1.48T + 343T^{2} \) |
| 11 | \( 1 + 15.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 55.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 46.4T + 4.91e3T^{2} \) |
| 23 | \( 1 + 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 14.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 65.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 361.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 329.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 208.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 435.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 114.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 270.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 266.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 811.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 748.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 57.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 267.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 494.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 508.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 103.T + 9.12e5T^{2} \) |
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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
−8.589805785196086534674499301675, −7.986251779733284716137431195555, −7.40119097784575022238489574885, −6.79824047910566559495377811249, −5.55915686599119254266702542188, −4.52100844854277642422811491797, −3.73287102160413414723395008237, −2.70510440512440433685749183131, −1.87382277149763875073925208826, −0.36609512959935230078116976809,
0.36609512959935230078116976809, 1.87382277149763875073925208826, 2.70510440512440433685749183131, 3.73287102160413414723395008237, 4.52100844854277642422811491797, 5.55915686599119254266702542188, 6.79824047910566559495377811249, 7.40119097784575022238489574885, 7.986251779733284716137431195555, 8.589805785196086534674499301675
