L(s) = 1 | + 2.90·3-s + 0.622·5-s + 1.52·7-s + 5.42·9-s + 1.09·11-s − 2.42·13-s + 1.80·15-s + 17-s + 5.80·19-s + 4.42·21-s − 8.57·23-s − 4.61·25-s + 7.05·27-s + 3.37·29-s + 3.33·31-s + 3.18·33-s + 0.949·35-s + 3.37·37-s − 7.05·39-s − 6.85·41-s − 7.05·43-s + 3.37·45-s + 1.24·47-s − 4.67·49-s + 2.90·51-s + 10.8·53-s + 0.682·55-s + ⋯ |
L(s) = 1 | + 1.67·3-s + 0.278·5-s + 0.576·7-s + 1.80·9-s + 0.330·11-s − 0.673·13-s + 0.466·15-s + 0.242·17-s + 1.33·19-s + 0.966·21-s − 1.78·23-s − 0.922·25-s + 1.35·27-s + 0.627·29-s + 0.598·31-s + 0.554·33-s + 0.160·35-s + 0.555·37-s − 1.12·39-s − 1.07·41-s − 1.07·43-s + 0.503·45-s + 0.181·47-s − 0.667·49-s + 0.406·51-s + 1.49·53-s + 0.0920·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.202170131\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.202170131\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 - 2.90T + 3T^{2} \) |
| 5 | \( 1 - 0.622T + 5T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 11 | \( 1 - 1.09T + 11T^{2} \) |
| 13 | \( 1 + 2.42T + 13T^{2} \) |
| 19 | \( 1 - 5.80T + 19T^{2} \) |
| 23 | \( 1 + 8.57T + 23T^{2} \) |
| 29 | \( 1 - 3.37T + 29T^{2} \) |
| 31 | \( 1 - 3.33T + 31T^{2} \) |
| 37 | \( 1 - 3.37T + 37T^{2} \) |
| 41 | \( 1 + 6.85T + 41T^{2} \) |
| 43 | \( 1 + 7.05T + 43T^{2} \) |
| 47 | \( 1 - 1.24T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 4.56T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 7.05T + 83T^{2} \) |
| 89 | \( 1 - 7.67T + 89T^{2} \) |
| 97 | \( 1 - 5.61T + 97T^{2} \) |
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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
−9.893607604999863987671115335258, −9.010350831722237756228982483094, −8.105270966734668183496840016528, −7.76525700013728849721458996541, −6.75956595201397050940332786690, −5.50910094680845334148731227198, −4.39588836077924870443032530560, −3.50113669657681810665516521481, −2.48983200319786744412653546615, −1.57628023352501225319203389283,
1.57628023352501225319203389283, 2.48983200319786744412653546615, 3.50113669657681810665516521481, 4.39588836077924870443032530560, 5.50910094680845334148731227198, 6.75956595201397050940332786690, 7.76525700013728849721458996541, 8.105270966734668183496840016528, 9.010350831722237756228982483094, 9.893607604999863987671115335258