L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·5-s + 2·6-s − 7-s − 8-s + 9-s − 2·10-s + 4·11-s − 2·12-s + 2·13-s + 14-s − 4·15-s + 16-s − 18-s + 6·19-s + 2·20-s + 2·21-s − 4·22-s + 23-s + 2·24-s − 25-s − 2·26-s + 4·27-s − 28-s + 4·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.577·12-s + 0.554·13-s + 0.267·14-s − 1.03·15-s + 1/4·16-s − 0.235·18-s + 1.37·19-s + 0.447·20-s + 0.436·21-s − 0.852·22-s + 0.208·23-s + 0.408·24-s − 1/5·25-s − 0.392·26-s + 0.769·27-s − 0.188·28-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93058 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.165505481\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.165505481\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.98146668797855, −13.17107551554964, −12.85866986827634, −12.10661955896789, −11.68859381958844, −11.48995478544696, −10.79347936770802, −10.42215894752268, −9.789240661003470, −9.453274055937247, −9.007735006391055, −8.476883858808427, −7.706258440025943, −7.119970560387393, −6.604818271087886, −6.192093780271551, −5.801177666515502, −5.263995873297917, −4.709311398001852, −3.743622160825724, −3.317474205816516, −2.506041546546641, −1.575899068635131, −1.277931053768455, −0.4461746401343343,
0.4461746401343343, 1.277931053768455, 1.575899068635131, 2.506041546546641, 3.317474205816516, 3.743622160825724, 4.709311398001852, 5.263995873297917, 5.801177666515502, 6.192093780271551, 6.604818271087886, 7.119970560387393, 7.706258440025943, 8.476883858808427, 9.007735006391055, 9.453274055937247, 9.789240661003470, 10.42215894752268, 10.79347936770802, 11.48995478544696, 11.68859381958844, 12.10661955896789, 12.85866986827634, 13.17107551554964, 13.98146668797855