L(s) = 1 | − 2-s − 2.80·3-s + 4-s − 0.554·5-s + 2.80·6-s − 7-s − 8-s + 4.85·9-s + 0.554·10-s + 5.24·11-s − 2.80·12-s + 14-s + 1.55·15-s + 16-s + 7.23·17-s − 4.85·18-s + 5.80·19-s − 0.554·20-s + 2.80·21-s − 5.24·22-s − 5.15·23-s + 2.80·24-s − 4.69·25-s − 5.18·27-s − 28-s + 5.43·29-s − 1.55·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.61·3-s + 0.5·4-s − 0.248·5-s + 1.14·6-s − 0.377·7-s − 0.353·8-s + 1.61·9-s + 0.175·10-s + 1.58·11-s − 0.808·12-s + 0.267·14-s + 0.401·15-s + 0.250·16-s + 1.75·17-s − 1.14·18-s + 1.33·19-s − 0.124·20-s + 0.611·21-s − 1.11·22-s − 1.07·23-s + 0.571·24-s − 0.938·25-s − 0.998·27-s − 0.188·28-s + 1.00·29-s − 0.283·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7137785335\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7137785335\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 + 0.554T + 5T^{2} \) |
| 11 | \( 1 - 5.24T + 11T^{2} \) |
| 17 | \( 1 - 7.23T + 17T^{2} \) |
| 19 | \( 1 - 5.80T + 19T^{2} \) |
| 23 | \( 1 + 5.15T + 23T^{2} \) |
| 29 | \( 1 - 5.43T + 29T^{2} \) |
| 31 | \( 1 + 5.54T + 31T^{2} \) |
| 37 | \( 1 - 0.335T + 37T^{2} \) |
| 41 | \( 1 - 3.11T + 41T^{2} \) |
| 43 | \( 1 + 6.91T + 43T^{2} \) |
| 47 | \( 1 - 5.56T + 47T^{2} \) |
| 53 | \( 1 + 7.78T + 53T^{2} \) |
| 59 | \( 1 - 7.24T + 59T^{2} \) |
| 61 | \( 1 - 8.45T + 61T^{2} \) |
| 67 | \( 1 - 2.59T + 67T^{2} \) |
| 71 | \( 1 + 9.18T + 71T^{2} \) |
| 73 | \( 1 - 4.29T + 73T^{2} \) |
| 79 | \( 1 + 4.00T + 79T^{2} \) |
| 83 | \( 1 + 8.36T + 83T^{2} \) |
| 89 | \( 1 + 2.11T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−9.227305851914362346773395236243, −8.084016970868927239143484149566, −7.32114186186231863650743219982, −6.63849992090276419279178304157, −5.89676381345442864684509794807, −5.38996600373743025591202846737, −4.14562977002813843987151646640, −3.34112019806461980728366281902, −1.54644520637227473966373298927, −0.71178262046978744835904803511,
0.71178262046978744835904803511, 1.54644520637227473966373298927, 3.34112019806461980728366281902, 4.14562977002813843987151646640, 5.38996600373743025591202846737, 5.89676381345442864684509794807, 6.63849992090276419279178304157, 7.32114186186231863650743219982, 8.084016970868927239143484149566, 9.227305851914362346773395236243