L(s) = 1 | − 3-s + 2.53·5-s − 3.22·7-s + 9-s + 3.10·11-s + 6.06·13-s − 2.53·15-s − 5.94·17-s + 3.22·21-s − 6.71·23-s + 1.41·25-s − 27-s − 7.12·29-s − 7.63·31-s − 3.10·33-s − 8.17·35-s − 2.10·37-s − 6.06·39-s + 7.04·41-s − 9.36·43-s + 2.53·45-s − 4.65·47-s + 3.41·49-s + 5.94·51-s + 6.35·53-s + 7.86·55-s + 0.268·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.13·5-s − 1.21·7-s + 0.333·9-s + 0.936·11-s + 1.68·13-s − 0.653·15-s − 1.44·17-s + 0.704·21-s − 1.40·23-s + 0.282·25-s − 0.192·27-s − 1.32·29-s − 1.37·31-s − 0.540·33-s − 1.38·35-s − 0.346·37-s − 0.971·39-s + 1.09·41-s − 1.42·43-s + 0.377·45-s − 0.678·47-s + 0.487·49-s + 0.832·51-s + 0.872·53-s + 1.06·55-s + 0.0349·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 - 2.53T + 5T^{2} \) |
| 7 | \( 1 + 3.22T + 7T^{2} \) |
| 11 | \( 1 - 3.10T + 11T^{2} \) |
| 13 | \( 1 - 6.06T + 13T^{2} \) |
| 17 | \( 1 + 5.94T + 17T^{2} \) |
| 23 | \( 1 + 6.71T + 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 + 7.63T + 31T^{2} \) |
| 37 | \( 1 + 2.10T + 37T^{2} \) |
| 41 | \( 1 - 7.04T + 41T^{2} \) |
| 43 | \( 1 + 9.36T + 43T^{2} \) |
| 47 | \( 1 + 4.65T + 47T^{2} \) |
| 53 | \( 1 - 6.35T + 53T^{2} \) |
| 59 | \( 1 - 0.268T + 59T^{2} \) |
| 61 | \( 1 - 1.58T + 61T^{2} \) |
| 67 | \( 1 - 1.30T + 67T^{2} \) |
| 71 | \( 1 - 8.27T + 71T^{2} \) |
| 73 | \( 1 - 0.921T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 - 0.0864T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 + 3.32T + 97T^{2} \) |
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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
−8.078902690652258827458392139714, −6.83285249484557034417181778694, −6.46045887925025699648286667198, −5.97652600098494953096189600176, −5.33168069060301270164346355614, −3.95492643007463403532496750408, −3.66746488726315519927808145276, −2.22197934432716306258600191803, −1.47862857242391269173572821545, 0,
1.47862857242391269173572821545, 2.22197934432716306258600191803, 3.66746488726315519927808145276, 3.95492643007463403532496750408, 5.33168069060301270164346355614, 5.97652600098494953096189600176, 6.46045887925025699648286667198, 6.83285249484557034417181778694, 8.078902690652258827458392139714