| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 6·11-s + 12-s + 6·13-s + 16-s + 18-s + 4·19-s + 6·22-s + 24-s + 6·26-s + 27-s − 8·29-s − 2·31-s + 32-s + 6·33-s + 36-s − 4·37-s + 4·38-s + 6·39-s + 10·41-s + 6·43-s + 6·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.288·12-s + 1.66·13-s + 1/4·16-s + 0.235·18-s + 0.917·19-s + 1.27·22-s + 0.204·24-s + 1.17·26-s + 0.192·27-s − 1.48·29-s − 0.359·31-s + 0.176·32-s + 1.04·33-s + 1/6·36-s − 0.657·37-s + 0.648·38-s + 0.960·39-s + 1.56·41-s + 0.914·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.405218277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.405218277\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.70382500895079284752277278128, −7.25944780439069805654356367225, −6.19830993809072437228776793264, −6.09786903102734751210552954741, −5.00311635053311657054331730697, −3.96445947083489824405417309038, −3.77190458275310488362840581754, −2.97554712932425145944275046918, −1.73733240194017929971528812815, −1.17746235306321458973551655578,
1.17746235306321458973551655578, 1.73733240194017929971528812815, 2.97554712932425145944275046918, 3.77190458275310488362840581754, 3.96445947083489824405417309038, 5.00311635053311657054331730697, 6.09786903102734751210552954741, 6.19830993809072437228776793264, 7.25944780439069805654356367225, 7.70382500895079284752277278128
