L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 11-s + 12-s + 1.41·13-s + 16-s − 4·17-s − 18-s − 2.58·19-s − 22-s − 3.65·23-s − 24-s − 5·25-s − 1.41·26-s + 27-s + 5.65·29-s − 11.0·31-s − 32-s + 33-s + 4·34-s + 36-s − 7.65·37-s + 2.58·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.301·11-s + 0.288·12-s + 0.392·13-s + 0.250·16-s − 0.970·17-s − 0.235·18-s − 0.593·19-s − 0.213·22-s − 0.762·23-s − 0.204·24-s − 25-s − 0.277·26-s + 0.192·27-s + 1.05·29-s − 1.98·31-s − 0.176·32-s + 0.174·33-s + 0.685·34-s + 0.166·36-s − 1.25·37-s + 0.419·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 + 11.0T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 1.65T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 - 7.65T + 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 2.34T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 9.65T + 79T^{2} \) |
| 83 | \( 1 - 0.928T + 83T^{2} \) |
| 89 | \( 1 + 5.41T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−8.546207677958704399328037653021, −7.66515287981542189023396363562, −6.89803630701357608047630337438, −6.28201946977882066944995791577, −5.29945299814021959004927834765, −4.15363615994975899704543567273, −3.48880534897610477124931807098, −2.29830846019740276361077734996, −1.61175161727465132907500839358, 0,
1.61175161727465132907500839358, 2.29830846019740276361077734996, 3.48880534897610477124931807098, 4.15363615994975899704543567273, 5.29945299814021959004927834765, 6.28201946977882066944995791577, 6.89803630701357608047630337438, 7.66515287981542189023396363562, 8.546207677958704399328037653021