| L(s) = 1 | + 1.41·3-s + 5-s − 7-s − 0.999·9-s − 3.41·11-s + 3.82·13-s + 1.41·15-s − 1.41·17-s + 19-s − 1.41·21-s − 9.07·23-s + 25-s − 5.65·27-s − 7.24·29-s − 2.41·31-s − 4.82·33-s − 35-s − 6.24·37-s + 5.41·39-s + 3.82·41-s − 43-s − 0.999·45-s + 7.07·47-s − 6·49-s − 2.00·51-s + 5.65·53-s − 3.41·55-s + ⋯ |
| L(s) = 1 | + 0.816·3-s + 0.447·5-s − 0.377·7-s − 0.333·9-s − 1.02·11-s + 1.06·13-s + 0.365·15-s − 0.342·17-s + 0.229·19-s − 0.308·21-s − 1.89·23-s + 0.200·25-s − 1.08·27-s − 1.34·29-s − 0.433·31-s − 0.840·33-s − 0.169·35-s − 1.02·37-s + 0.866·39-s + 0.597·41-s − 0.152·43-s − 0.149·45-s + 1.03·47-s − 0.857·49-s − 0.280·51-s + 0.777·53-s − 0.460·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 9.07T + 23T^{2} \) |
| 29 | \( 1 + 7.24T + 29T^{2} \) |
| 31 | \( 1 + 2.41T + 31T^{2} \) |
| 37 | \( 1 + 6.24T + 37T^{2} \) |
| 41 | \( 1 - 3.82T + 41T^{2} \) |
| 47 | \( 1 - 7.07T + 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 - 6.82T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 + 7.24T + 67T^{2} \) |
| 71 | \( 1 - 7.89T + 71T^{2} \) |
| 73 | \( 1 + 0.757T + 73T^{2} \) |
| 79 | \( 1 - 5.58T + 79T^{2} \) |
| 83 | \( 1 - 7.65T + 83T^{2} \) |
| 89 | \( 1 - 5.07T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−8.133708916736079930231012043185, −7.78556307794424872931323970899, −6.73296549020901483391347047688, −5.83123233886137324021761831204, −5.44000528220951560458101990190, −4.08936975374480196404145338967, −3.43516904386278260256600510364, −2.52070728159632492265339737755, −1.76940601242275908314796476994, 0,
1.76940601242275908314796476994, 2.52070728159632492265339737755, 3.43516904386278260256600510364, 4.08936975374480196404145338967, 5.44000528220951560458101990190, 5.83123233886137324021761831204, 6.73296549020901483391347047688, 7.78556307794424872931323970899, 8.133708916736079930231012043185
