| L(s) = 1 | + 2.27·3-s − 1.59·5-s + 0.209·7-s + 2.19·9-s − 3.58·13-s − 3.64·15-s + 3.14·17-s − 19-s + 0.476·21-s + 3.39·23-s − 2.44·25-s − 1.82·27-s + 7.67·29-s − 0.788·31-s − 0.333·35-s − 5.39·37-s − 8.18·39-s + 4.42·41-s − 9.42·43-s − 3.50·45-s + 0.507·47-s − 6.95·49-s + 7.16·51-s − 9.23·53-s − 2.27·57-s − 12.2·59-s + 2.39·61-s + ⋯ |
| L(s) = 1 | + 1.31·3-s − 0.714·5-s + 0.0790·7-s + 0.732·9-s − 0.995·13-s − 0.940·15-s + 0.761·17-s − 0.229·19-s + 0.103·21-s + 0.706·23-s − 0.489·25-s − 0.352·27-s + 1.42·29-s − 0.141·31-s − 0.0564·35-s − 0.887·37-s − 1.31·39-s + 0.691·41-s − 1.43·43-s − 0.523·45-s + 0.0739·47-s − 0.993·49-s + 1.00·51-s − 1.26·53-s − 0.301·57-s − 1.59·59-s + 0.307·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.27T + 3T^{2} \) |
| 5 | \( 1 + 1.59T + 5T^{2} \) |
| 7 | \( 1 - 0.209T + 7T^{2} \) |
| 13 | \( 1 + 3.58T + 13T^{2} \) |
| 17 | \( 1 - 3.14T + 17T^{2} \) |
| 23 | \( 1 - 3.39T + 23T^{2} \) |
| 29 | \( 1 - 7.67T + 29T^{2} \) |
| 31 | \( 1 + 0.788T + 31T^{2} \) |
| 37 | \( 1 + 5.39T + 37T^{2} \) |
| 41 | \( 1 - 4.42T + 41T^{2} \) |
| 43 | \( 1 + 9.42T + 43T^{2} \) |
| 47 | \( 1 - 0.507T + 47T^{2} \) |
| 53 | \( 1 + 9.23T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 2.39T + 61T^{2} \) |
| 67 | \( 1 - 8.20T + 67T^{2} \) |
| 71 | \( 1 + 0.166T + 71T^{2} \) |
| 73 | \( 1 - 4.16T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 7.36T + 83T^{2} \) |
| 89 | \( 1 - 3.16T + 89T^{2} \) |
| 97 | \( 1 - 6.67T + 97T^{2} \) |
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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
−7.54517465609515354353157382103, −6.98732590704447220807494594968, −6.11393669976794675857105919438, −5.05487588839277379686141229968, −4.54186819882585040658309579976, −3.57540371926977132010705112424, −3.13099584568387214834805502450, −2.37348490085986364075785207367, −1.42136042907514209726162402355, 0,
1.42136042907514209726162402355, 2.37348490085986364075785207367, 3.13099584568387214834805502450, 3.57540371926977132010705112424, 4.54186819882585040658309579976, 5.05487588839277379686141229968, 6.11393669976794675857105919438, 6.98732590704447220807494594968, 7.54517465609515354353157382103
