| L(s) = 1 | − 1.11·3-s + 2.58·5-s + 0.902·7-s − 1.74·9-s + 4.37·11-s − 0.575·13-s − 2.89·15-s − 0.596·17-s − 2.17·19-s − 1.01·21-s − 2.92·23-s + 1.68·25-s + 5.31·27-s + 7.59·29-s − 6.25·31-s − 4.90·33-s + 2.33·35-s − 1.33·37-s + 0.643·39-s + 1.89·41-s + 12.0·43-s − 4.51·45-s + 3.13·47-s − 6.18·49-s + 0.668·51-s + 7.54·53-s + 11.3·55-s + ⋯ |
| L(s) = 1 | − 0.646·3-s + 1.15·5-s + 0.341·7-s − 0.582·9-s + 1.31·11-s − 0.159·13-s − 0.747·15-s − 0.144·17-s − 0.498·19-s − 0.220·21-s − 0.610·23-s + 0.337·25-s + 1.02·27-s + 1.41·29-s − 1.12·31-s − 0.852·33-s + 0.394·35-s − 0.218·37-s + 0.103·39-s + 0.295·41-s + 1.83·43-s − 0.673·45-s + 0.457·47-s − 0.883·49-s + 0.0935·51-s + 1.03·53-s + 1.52·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.053316732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.053316732\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 + 1.11T + 3T^{2} \) |
| 5 | \( 1 - 2.58T + 5T^{2} \) |
| 7 | \( 1 - 0.902T + 7T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 13 | \( 1 + 0.575T + 13T^{2} \) |
| 17 | \( 1 + 0.596T + 17T^{2} \) |
| 19 | \( 1 + 2.17T + 19T^{2} \) |
| 23 | \( 1 + 2.92T + 23T^{2} \) |
| 29 | \( 1 - 7.59T + 29T^{2} \) |
| 31 | \( 1 + 6.25T + 31T^{2} \) |
| 37 | \( 1 + 1.33T + 37T^{2} \) |
| 41 | \( 1 - 1.89T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 - 3.13T + 47T^{2} \) |
| 53 | \( 1 - 7.54T + 53T^{2} \) |
| 59 | \( 1 - 3.10T + 59T^{2} \) |
| 61 | \( 1 + 6.45T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 0.0385T + 71T^{2} \) |
| 73 | \( 1 + 8.42T + 73T^{2} \) |
| 79 | \( 1 - 6.34T + 79T^{2} \) |
| 83 | \( 1 + 0.391T + 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−8.178557348887472958045786062389, −7.17895996455989072803281090282, −6.39841017276139703680418106441, −6.03083096729491887958718497677, −5.37216808995673798712398772607, −4.57080758045624627698559039071, −3.75115204966267463072240268289, −2.57513818431749908728868136053, −1.82924096573443025836799927865, −0.795416765880428250894609864545,
0.795416765880428250894609864545, 1.82924096573443025836799927865, 2.57513818431749908728868136053, 3.75115204966267463072240268289, 4.57080758045624627698559039071, 5.37216808995673798712398772607, 6.03083096729491887958718497677, 6.39841017276139703680418106441, 7.17895996455989072803281090282, 8.178557348887472958045786062389
