L(s) = 1 | − 3-s + 1.93·5-s + 2.74·7-s + 9-s − 0.761·11-s − 0.823·13-s − 1.93·15-s − 3.77·17-s + 2.01·19-s − 2.74·21-s − 7.45·23-s − 1.24·25-s − 27-s − 5.07·29-s + 10.0·31-s + 0.761·33-s + 5.31·35-s + 9.29·37-s + 0.823·39-s − 9.81·41-s − 11.3·43-s + 1.93·45-s − 12.1·47-s + 0.535·49-s + 3.77·51-s + 0.223·53-s − 1.47·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.866·5-s + 1.03·7-s + 0.333·9-s − 0.229·11-s − 0.228·13-s − 0.500·15-s − 0.916·17-s + 0.461·19-s − 0.599·21-s − 1.55·23-s − 0.249·25-s − 0.192·27-s − 0.942·29-s + 1.80·31-s + 0.132·33-s + 0.899·35-s + 1.52·37-s + 0.131·39-s − 1.53·41-s − 1.73·43-s + 0.288·45-s − 1.77·47-s + 0.0764·49-s + 0.529·51-s + 0.0307·53-s − 0.198·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 251 | \( 1 + T \) |
good | 5 | \( 1 - 1.93T + 5T^{2} \) |
| 7 | \( 1 - 2.74T + 7T^{2} \) |
| 11 | \( 1 + 0.761T + 11T^{2} \) |
| 13 | \( 1 + 0.823T + 13T^{2} \) |
| 17 | \( 1 + 3.77T + 17T^{2} \) |
| 19 | \( 1 - 2.01T + 19T^{2} \) |
| 23 | \( 1 + 7.45T + 23T^{2} \) |
| 29 | \( 1 + 5.07T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 9.29T + 37T^{2} \) |
| 41 | \( 1 + 9.81T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 - 0.223T + 53T^{2} \) |
| 59 | \( 1 + 6.47T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 5.53T + 67T^{2} \) |
| 71 | \( 1 - 1.82T + 71T^{2} \) |
| 73 | \( 1 + 5.35T + 73T^{2} \) |
| 79 | \( 1 - 4.27T + 79T^{2} \) |
| 83 | \( 1 + 1.53T + 83T^{2} \) |
| 89 | \( 1 + 0.918T + 89T^{2} \) |
| 97 | \( 1 + 6.91T + 97T^{2} \) |
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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
−7.989643299603307420909493044920, −6.84090224364660709745171233479, −6.23033145807018119891808582598, −5.62289892771754236637959378227, −4.79870931255452825284692796585, −4.40542484997865032579559388418, −3.14528971643114779807072377102, −2.02326782252240873751244713594, −1.53961263304277966937764343922, 0,
1.53961263304277966937764343922, 2.02326782252240873751244713594, 3.14528971643114779807072377102, 4.40542484997865032579559388418, 4.79870931255452825284692796585, 5.62289892771754236637959378227, 6.23033145807018119891808582598, 6.84090224364660709745171233479, 7.989643299603307420909493044920