| L(s) = 1 | + 3-s − 3.49·5-s + 4.58·7-s + 9-s + 0.512·11-s − 0.325·13-s − 3.49·15-s − 2.49·17-s + 3.26·19-s + 4.58·21-s − 4.00·23-s + 7.20·25-s + 27-s + 1.72·29-s + 5.79·31-s + 0.512·33-s − 16.0·35-s + 7.03·37-s − 0.325·39-s + 1.69·41-s − 7.20·43-s − 3.49·45-s + 2.49·47-s + 13.9·49-s − 2.49·51-s + 2.26·53-s − 1.79·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.56·5-s + 1.73·7-s + 0.333·9-s + 0.154·11-s − 0.0903·13-s − 0.902·15-s − 0.604·17-s + 0.748·19-s + 0.999·21-s − 0.835·23-s + 1.44·25-s + 0.192·27-s + 0.320·29-s + 1.04·31-s + 0.0892·33-s − 2.70·35-s + 1.15·37-s − 0.0521·39-s + 0.264·41-s − 1.09·43-s − 0.520·45-s + 0.363·47-s + 1.99·49-s − 0.348·51-s + 0.311·53-s − 0.241·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.188343758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.188343758\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 3.49T + 5T^{2} \) |
| 7 | \( 1 - 4.58T + 7T^{2} \) |
| 11 | \( 1 - 0.512T + 11T^{2} \) |
| 13 | \( 1 + 0.325T + 13T^{2} \) |
| 17 | \( 1 + 2.49T + 17T^{2} \) |
| 19 | \( 1 - 3.26T + 19T^{2} \) |
| 23 | \( 1 + 4.00T + 23T^{2} \) |
| 29 | \( 1 - 1.72T + 29T^{2} \) |
| 31 | \( 1 - 5.79T + 31T^{2} \) |
| 37 | \( 1 - 7.03T + 37T^{2} \) |
| 41 | \( 1 - 1.69T + 41T^{2} \) |
| 43 | \( 1 + 7.20T + 43T^{2} \) |
| 47 | \( 1 - 2.49T + 47T^{2} \) |
| 53 | \( 1 - 2.26T + 53T^{2} \) |
| 59 | \( 1 + 6.76T + 59T^{2} \) |
| 61 | \( 1 + 1.57T + 61T^{2} \) |
| 67 | \( 1 - 0.691T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 3.41T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 3.12T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 8.68T + 97T^{2} \) |
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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
−8.275882514209447165354293198353, −7.82120678160411716862227818125, −7.38097750240993845002788003482, −6.39582623992654996617191617647, −5.16007325984587631042295133015, −4.48041503400021122149954588325, −4.02841736362897477494913093650, −3.02425665992598589853575493818, −1.97854773439203712344921041907, −0.853361803869348927451257825855,
0.853361803869348927451257825855, 1.97854773439203712344921041907, 3.02425665992598589853575493818, 4.02841736362897477494913093650, 4.48041503400021122149954588325, 5.16007325984587631042295133015, 6.39582623992654996617191617647, 7.38097750240993845002788003482, 7.82120678160411716862227818125, 8.275882514209447165354293198353
