| L(s) = 1 | − 3-s − 0.395·7-s + 9-s + 2.79·11-s − 5.70·13-s − 4.25·17-s − 6.10·19-s + 0.395·21-s + 6.36·23-s − 27-s + 7.96·29-s + 31-s − 2.79·33-s + 9.80·37-s + 5.70·39-s + 3.20·41-s + 3.31·43-s − 7.05·47-s − 6.84·49-s + 4.25·51-s + 1.05·53-s + 6.10·57-s + 3.34·59-s + 12.6·61-s − 0.395·63-s − 13.8·67-s − 6.36·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.149·7-s + 0.333·9-s + 0.841·11-s − 1.58·13-s − 1.03·17-s − 1.40·19-s + 0.0863·21-s + 1.32·23-s − 0.192·27-s + 1.47·29-s + 0.179·31-s − 0.486·33-s + 1.61·37-s + 0.913·39-s + 0.501·41-s + 0.504·43-s − 1.02·47-s − 0.977·49-s + 0.596·51-s + 0.144·53-s + 0.808·57-s + 0.435·59-s + 1.61·61-s − 0.0498·63-s − 1.68·67-s − 0.765·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 0.395T + 7T^{2} \) |
| 11 | \( 1 - 2.79T + 11T^{2} \) |
| 13 | \( 1 + 5.70T + 13T^{2} \) |
| 17 | \( 1 + 4.25T + 17T^{2} \) |
| 19 | \( 1 + 6.10T + 19T^{2} \) |
| 23 | \( 1 - 6.36T + 23T^{2} \) |
| 29 | \( 1 - 7.96T + 29T^{2} \) |
| 37 | \( 1 - 9.80T + 37T^{2} \) |
| 41 | \( 1 - 3.20T + 41T^{2} \) |
| 43 | \( 1 - 3.31T + 43T^{2} \) |
| 47 | \( 1 + 7.05T + 47T^{2} \) |
| 53 | \( 1 - 1.05T + 53T^{2} \) |
| 59 | \( 1 - 3.34T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 0.156T + 79T^{2} \) |
| 83 | \( 1 + 7.74T + 83T^{2} \) |
| 89 | \( 1 - 7.55T + 89T^{2} \) |
| 97 | \( 1 + 4.79T + 97T^{2} \) |
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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
−7.16509753729126704060703714798, −6.60488455285667833541640199265, −6.24672863581393780367318530490, −5.17591795704975992049807379306, −4.56650055600923414777920082755, −4.14186024962236756116313895870, −2.87050877325242369540849083642, −2.26656936942966263329427527368, −1.09079696923655634106944271913, 0,
1.09079696923655634106944271913, 2.26656936942966263329427527368, 2.87050877325242369540849083642, 4.14186024962236756116313895870, 4.56650055600923414777920082755, 5.17591795704975992049807379306, 6.24672863581393780367318530490, 6.60488455285667833541640199265, 7.16509753729126704060703714798
