| L(s) = 1 | + 3.21·3-s + 2.59·7-s + 7.35·9-s − 0.741·11-s + 3.78·13-s + 3.16·17-s + 19-s + 8.35·21-s + 0.570·23-s + 14.0·27-s + 6·29-s − 5.83·31-s − 2.38·33-s + 1.40·37-s + 12.1·39-s − 3.83·41-s − 2.59·43-s + 5.08·47-s − 0.258·49-s + 10.1·51-s + 0.160·53-s + 3.21·57-s − 8.35·59-s − 8.57·61-s + 19.0·63-s − 14.8·67-s + 1.83·69-s + ⋯ |
| L(s) = 1 | + 1.85·3-s + 0.981·7-s + 2.45·9-s − 0.223·11-s + 1.05·13-s + 0.768·17-s + 0.229·19-s + 1.82·21-s + 0.119·23-s + 2.69·27-s + 1.11·29-s − 1.04·31-s − 0.415·33-s + 0.230·37-s + 1.95·39-s − 0.599·41-s − 0.395·43-s + 0.741·47-s − 0.0369·49-s + 1.42·51-s + 0.0221·53-s + 0.426·57-s − 1.08·59-s − 1.09·61-s + 2.40·63-s − 1.81·67-s + 0.221·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.570103475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.570103475\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 3.21T + 3T^{2} \) |
| 7 | \( 1 - 2.59T + 7T^{2} \) |
| 11 | \( 1 + 0.741T + 11T^{2} \) |
| 13 | \( 1 - 3.78T + 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 23 | \( 1 - 0.570T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 5.83T + 31T^{2} \) |
| 37 | \( 1 - 1.40T + 37T^{2} \) |
| 41 | \( 1 + 3.83T + 41T^{2} \) |
| 43 | \( 1 + 2.59T + 43T^{2} \) |
| 47 | \( 1 - 5.08T + 47T^{2} \) |
| 53 | \( 1 - 0.160T + 53T^{2} \) |
| 59 | \( 1 + 8.35T + 59T^{2} \) |
| 61 | \( 1 + 8.57T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 + 3.64T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 1.83T + 79T^{2} \) |
| 83 | \( 1 + 4.19T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 - 3.78T + 97T^{2} \) |
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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
−7.913941798417312901950722066288, −7.56199433405950052246282382080, −6.73901923356151045237493484106, −5.75873185293318588080892296265, −4.82160934863106253555086092454, −4.16288047660084640371526783237, −3.36500658600500571994941914855, −2.82954574624420414236595016606, −1.78423777392927087145454219099, −1.25308871688910995531677554692,
1.25308871688910995531677554692, 1.78423777392927087145454219099, 2.82954574624420414236595016606, 3.36500658600500571994941914855, 4.16288047660084640371526783237, 4.82160934863106253555086092454, 5.75873185293318588080892296265, 6.73901923356151045237493484106, 7.56199433405950052246282382080, 7.913941798417312901950722066288
