L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 4·7-s − 8-s + 9-s − 6·11-s − 2·12-s − 2·13-s + 4·14-s + 16-s − 17-s − 18-s − 4·19-s + 8·21-s + 6·22-s + 2·24-s − 5·25-s + 2·26-s + 4·27-s − 4·28-s + 4·31-s − 32-s + 12·33-s + 34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 1.80·11-s − 0.577·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s + 1.74·21-s + 1.27·22-s + 0.408·24-s − 25-s + 0.392·26-s + 0.769·27-s − 0.755·28-s + 0.718·31-s − 0.176·32-s + 2.08·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.90008442701427, −13.34362251091140, −12.97819592754304, −12.54979946956442, −12.14081327234015, −11.57324164505906, −11.03635236220203, −10.57865621995593, −10.24466929102898, −9.742850427308515, −9.443552533192343, −8.597085462543592, −8.174312466940740, −7.593275209770476, −7.078466780380501, −6.492187858361894, −6.082581321046240, −5.746955808820688, −5.039474696413683, −4.582190019784711, −3.754313887492880, −2.936499770864329, −2.623649589388568, −1.946050999953261, −0.8011452024165587, 0, 0,
0.8011452024165587, 1.946050999953261, 2.623649589388568, 2.936499770864329, 3.754313887492880, 4.582190019784711, 5.039474696413683, 5.746955808820688, 6.082581321046240, 6.492187858361894, 7.078466780380501, 7.593275209770476, 8.174312466940740, 8.597085462543592, 9.443552533192343, 9.742850427308515, 10.24466929102898, 10.57865621995593, 11.03635236220203, 11.57324164505906, 12.14081327234015, 12.54979946956442, 12.97819592754304, 13.34362251091140, 13.90008442701427