| L(s) = 1 | + 3.17·3-s − 4.27·5-s − 4.73·7-s + 7.07·9-s + 3.63·11-s + 3.74·13-s − 13.5·15-s − 1.80·17-s − 19-s − 15.0·21-s − 2.48·23-s + 13.2·25-s + 12.9·27-s + 0.417·29-s − 6.42·31-s + 11.5·33-s + 20.2·35-s − 4.97·37-s + 11.8·39-s − 0.0129·41-s − 11.1·43-s − 30.2·45-s − 6.11·47-s + 15.4·49-s − 5.72·51-s + 53-s − 15.5·55-s + ⋯ |
| L(s) = 1 | + 1.83·3-s − 1.91·5-s − 1.78·7-s + 2.35·9-s + 1.09·11-s + 1.03·13-s − 3.50·15-s − 0.437·17-s − 0.229·19-s − 3.28·21-s − 0.518·23-s + 2.65·25-s + 2.49·27-s + 0.0774·29-s − 1.15·31-s + 2.00·33-s + 3.42·35-s − 0.818·37-s + 1.90·39-s − 0.00202·41-s − 1.70·43-s − 4.51·45-s − 0.892·47-s + 2.20·49-s − 0.802·51-s + 0.137·53-s − 2.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 - 3.17T + 3T^{2} \) |
| 5 | \( 1 + 4.27T + 5T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 - 3.63T + 11T^{2} \) |
| 13 | \( 1 - 3.74T + 13T^{2} \) |
| 17 | \( 1 + 1.80T + 17T^{2} \) |
| 23 | \( 1 + 2.48T + 23T^{2} \) |
| 29 | \( 1 - 0.417T + 29T^{2} \) |
| 31 | \( 1 + 6.42T + 31T^{2} \) |
| 37 | \( 1 + 4.97T + 37T^{2} \) |
| 41 | \( 1 + 0.0129T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 6.11T + 47T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 4.40T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 3.93T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 3.91T + 89T^{2} \) |
| 97 | \( 1 - 7.08T + 97T^{2} \) |
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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
−8.257788940175544047310464952898, −7.38552275071253152175348585419, −6.90469739539470310863966432608, −6.25824946631262215487455060890, −4.49834066949631912861439950509, −3.79910321016174721889732950299, −3.49394881427489071240017062017, −2.96872277978238233703950482808, −1.54097321056449223719233099180, 0,
1.54097321056449223719233099180, 2.96872277978238233703950482808, 3.49394881427489071240017062017, 3.79910321016174721889732950299, 4.49834066949631912861439950509, 6.25824946631262215487455060890, 6.90469739539470310863966432608, 7.38552275071253152175348585419, 8.257788940175544047310464952898
