L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 12-s − 2·13-s + 16-s + 17-s − 18-s + 4·19-s − 6·23-s + 24-s − 5·25-s + 2·26-s − 27-s + 10·31-s − 32-s − 34-s + 36-s + 8·37-s − 4·38-s + 2·39-s − 6·41-s − 4·43-s + 6·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s − 1.25·23-s + 0.204·24-s − 25-s + 0.392·26-s − 0.192·27-s + 1.79·31-s − 0.176·32-s − 0.171·34-s + 1/6·36-s + 1.31·37-s − 0.648·38-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.947184329792423035407392570167, −7.25305712045924996183772000512, −6.46045064507146826089961982136, −5.86386080475725522331193199391, −5.05078174394179622216561477864, −4.20559745353981447975579336986, −3.18505726016215617296720668342, −2.20095556111953042794406475709, −1.17658582231940763813732019251, 0,
1.17658582231940763813732019251, 2.20095556111953042794406475709, 3.18505726016215617296720668342, 4.20559745353981447975579336986, 5.05078174394179622216561477864, 5.86386080475725522331193199391, 6.46045064507146826089961982136, 7.25305712045924996183772000512, 7.947184329792423035407392570167