L(s) = 1 | − 3·3-s + 3·5-s + 7-s + 6·9-s − 4·11-s − 9·15-s + 2·17-s + 4·19-s − 3·21-s − 23-s + 4·25-s − 9·27-s + 2·29-s + 10·31-s + 12·33-s + 3·35-s − 4·37-s + 12·41-s + 18·45-s − 10·47-s + 49-s − 6·51-s + 4·53-s − 12·55-s − 12·57-s − 9·59-s − 13·61-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 1.34·5-s + 0.377·7-s + 2·9-s − 1.20·11-s − 2.32·15-s + 0.485·17-s + 0.917·19-s − 0.654·21-s − 0.208·23-s + 4/5·25-s − 1.73·27-s + 0.371·29-s + 1.79·31-s + 2.08·33-s + 0.507·35-s − 0.657·37-s + 1.87·41-s + 2.68·45-s − 1.45·47-s + 1/7·49-s − 0.840·51-s + 0.549·53-s − 1.61·55-s − 1.58·57-s − 1.17·59-s − 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.690462243\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.690462243\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.92003119914506, −13.38552922399839, −13.15758752352341, −12.40858728536818, −12.02775211468425, −11.64950779396917, −10.93257784718989, −10.46868293802067, −10.28924803991223, −9.696168701826081, −9.266499500231208, −8.428323398608091, −7.590586260024894, −7.513699149501322, −6.413990537064157, −6.273108715856679, −5.683907607781908, −5.286427896726107, −4.777506742984098, −4.408425484539104, −3.218399182297735, −2.634998261516120, −1.799519150234252, −1.209530027902054, −0.5197947927511812,
0.5197947927511812, 1.209530027902054, 1.799519150234252, 2.634998261516120, 3.218399182297735, 4.408425484539104, 4.777506742984098, 5.286427896726107, 5.683907607781908, 6.273108715856679, 6.413990537064157, 7.513699149501322, 7.590586260024894, 8.428323398608091, 9.266499500231208, 9.696168701826081, 10.28924803991223, 10.46868293802067, 10.93257784718989, 11.64950779396917, 12.02775211468425, 12.40858728536818, 13.15758752352341, 13.38552922399839, 13.92003119914506