L(s) = 1 | + 2.43·3-s + 9.42·5-s − 9.58·7-s − 21.0·9-s + 37.8·11-s + 30.7·13-s + 22.9·15-s + 35.3·17-s + 19·19-s − 23.3·21-s − 88.7·23-s − 36.1·25-s − 117.·27-s + 226.·29-s + 320.·31-s + 92.2·33-s − 90.3·35-s − 346.·37-s + 75.0·39-s − 150.·41-s + 284.·43-s − 198.·45-s + 240.·47-s − 251.·49-s + 86.1·51-s + 539.·53-s + 356.·55-s + ⋯ |
L(s) = 1 | + 0.469·3-s + 0.843·5-s − 0.517·7-s − 0.779·9-s + 1.03·11-s + 0.656·13-s + 0.395·15-s + 0.504·17-s + 0.229·19-s − 0.242·21-s − 0.804·23-s − 0.289·25-s − 0.835·27-s + 1.44·29-s + 1.85·31-s + 0.486·33-s − 0.436·35-s − 1.54·37-s + 0.308·39-s − 0.574·41-s + 1.00·43-s − 0.657·45-s + 0.746·47-s − 0.731·49-s + 0.236·51-s + 1.39·53-s + 0.875·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.977134014\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.977134014\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
good | 3 | \( 1 - 2.43T + 27T^{2} \) |
| 5 | \( 1 - 9.42T + 125T^{2} \) |
| 7 | \( 1 + 9.58T + 343T^{2} \) |
| 11 | \( 1 - 37.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 30.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 35.3T + 4.91e3T^{2} \) |
| 23 | \( 1 + 88.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 320.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 346.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 150.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 284.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 240.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 539.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 889.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 377.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 78.4T + 3.00e5T^{2} \) |
| 71 | \( 1 - 924.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 576.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 601.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 828.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 101.T + 9.12e5T^{2} \) |
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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
−9.360475189664118029243552745093, −8.622841838133930081553032723438, −7.933616282920292425087118749112, −6.57236547523576223978099824350, −6.19800699900193707421002178221, −5.24726987015075832363312503093, −3.94397570536716689802066036358, −3.10960139536546827488759717742, −2.09143196595837462915225098542, −0.882292532397910647050880118356,
0.882292532397910647050880118356, 2.09143196595837462915225098542, 3.10960139536546827488759717742, 3.94397570536716689802066036358, 5.24726987015075832363312503093, 6.19800699900193707421002178221, 6.57236547523576223978099824350, 7.933616282920292425087118749112, 8.622841838133930081553032723438, 9.360475189664118029243552745093