| L(s) = 1 | − 1.57·3-s − 4.10·5-s − 3.99·7-s − 0.525·9-s − 5.59·11-s + 0.327·13-s + 6.45·15-s + 1.21·17-s + 19-s + 6.28·21-s + 0.458·23-s + 11.8·25-s + 5.54·27-s − 2.71·29-s + 0.456·31-s + 8.79·33-s + 16.4·35-s + 0.0670·37-s − 0.514·39-s − 1.54·41-s + 8.95·43-s + 2.15·45-s − 0.548·47-s + 8.97·49-s − 1.91·51-s − 5.96·53-s + 22.9·55-s + ⋯ |
| L(s) = 1 | − 0.908·3-s − 1.83·5-s − 1.51·7-s − 0.175·9-s − 1.68·11-s + 0.0907·13-s + 1.66·15-s + 0.295·17-s + 0.229·19-s + 1.37·21-s + 0.0956·23-s + 2.36·25-s + 1.06·27-s − 0.503·29-s + 0.0819·31-s + 1.53·33-s + 2.77·35-s + 0.0110·37-s − 0.0823·39-s − 0.241·41-s + 1.36·43-s + 0.321·45-s − 0.0800·47-s + 1.28·49-s − 0.268·51-s − 0.819·53-s + 3.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 1.57T + 3T^{2} \) |
| 5 | \( 1 + 4.10T + 5T^{2} \) |
| 7 | \( 1 + 3.99T + 7T^{2} \) |
| 11 | \( 1 + 5.59T + 11T^{2} \) |
| 13 | \( 1 - 0.327T + 13T^{2} \) |
| 17 | \( 1 - 1.21T + 17T^{2} \) |
| 23 | \( 1 - 0.458T + 23T^{2} \) |
| 29 | \( 1 + 2.71T + 29T^{2} \) |
| 31 | \( 1 - 0.456T + 31T^{2} \) |
| 37 | \( 1 - 0.0670T + 37T^{2} \) |
| 41 | \( 1 + 1.54T + 41T^{2} \) |
| 43 | \( 1 - 8.95T + 43T^{2} \) |
| 47 | \( 1 + 0.548T + 47T^{2} \) |
| 53 | \( 1 + 5.96T + 53T^{2} \) |
| 59 | \( 1 - 4.29T + 59T^{2} \) |
| 61 | \( 1 - 1.43T + 61T^{2} \) |
| 67 | \( 1 - 2.85T + 67T^{2} \) |
| 71 | \( 1 - 5.48T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 83 | \( 1 - 3.71T + 83T^{2} \) |
| 89 | \( 1 + 3.84T + 89T^{2} \) |
| 97 | \( 1 + 6.85T + 97T^{2} \) |
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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
−7.61256706030954481442535054219, −7.11189391249999223573382227221, −6.29847199381970044430036357365, −5.57255172962044953017317220482, −4.89963438122432598399717005885, −4.01664926379166001469881009471, −3.21957140877766600912776218144, −2.72253757289936569101546788667, −0.64776199887710157436986602223, 0,
0.64776199887710157436986602223, 2.72253757289936569101546788667, 3.21957140877766600912776218144, 4.01664926379166001469881009471, 4.89963438122432598399717005885, 5.57255172962044953017317220482, 6.29847199381970044430036357365, 7.11189391249999223573382227221, 7.61256706030954481442535054219
