| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 3·11-s − 13-s − 15-s − 3·17-s + 6·19-s − 21-s + 23-s + 25-s + 27-s + 8·29-s − 4·31-s − 3·33-s + 35-s + 5·37-s − 39-s − 5·41-s − 6·43-s − 45-s + 8·47-s − 6·49-s − 3·51-s + 9·53-s + 3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s − 0.258·15-s − 0.727·17-s + 1.37·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.48·29-s − 0.718·31-s − 0.522·33-s + 0.169·35-s + 0.821·37-s − 0.160·39-s − 0.780·41-s − 0.914·43-s − 0.149·45-s + 1.16·47-s − 6/7·49-s − 0.420·51-s + 1.23·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.74405341227325566994858832264, −7.12002842881362669504853749566, −6.43908583419440290766519782358, −5.42500779361973359107038425283, −4.78673533954818868051209963936, −3.96704610570725328498128864900, −3.03325237604964211977491404431, −2.61209375157114139989026852175, −1.32892721737441945891048437470, 0,
1.32892721737441945891048437470, 2.61209375157114139989026852175, 3.03325237604964211977491404431, 3.96704610570725328498128864900, 4.78673533954818868051209963936, 5.42500779361973359107038425283, 6.43908583419440290766519782358, 7.12002842881362669504853749566, 7.74405341227325566994858832264
