| L(s) = 1 | + 2-s + 2.44·3-s + 4-s + 2.44·6-s − 0.0417·7-s + 8-s + 2.98·9-s − 2.50·11-s + 2.44·12-s − 0.0417·14-s + 16-s + 6.16·17-s + 2.98·18-s − 4.90·19-s − 0.102·21-s − 2.50·22-s + 7.48·23-s + 2.44·24-s − 0.0379·27-s − 0.0417·28-s + 2.24·29-s − 0.913·31-s + 32-s − 6.11·33-s + 6.16·34-s + 2.98·36-s + 5.58·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.41·3-s + 0.5·4-s + 0.998·6-s − 0.0157·7-s + 0.353·8-s + 0.994·9-s − 0.753·11-s + 0.706·12-s − 0.0111·14-s + 0.250·16-s + 1.49·17-s + 0.703·18-s − 1.12·19-s − 0.0222·21-s − 0.533·22-s + 1.55·23-s + 0.499·24-s − 0.00729·27-s − 0.00789·28-s + 0.416·29-s − 0.164·31-s + 0.176·32-s − 1.06·33-s + 1.05·34-s + 0.497·36-s + 0.918·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.868158677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.868158677\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 7 | \( 1 + 0.0417T + 7T^{2} \) |
| 11 | \( 1 + 2.50T + 11T^{2} \) |
| 17 | \( 1 - 6.16T + 17T^{2} \) |
| 19 | \( 1 + 4.90T + 19T^{2} \) |
| 23 | \( 1 - 7.48T + 23T^{2} \) |
| 29 | \( 1 - 2.24T + 29T^{2} \) |
| 31 | \( 1 + 0.913T + 31T^{2} \) |
| 37 | \( 1 - 5.58T + 37T^{2} \) |
| 41 | \( 1 + 8.34T + 41T^{2} \) |
| 43 | \( 1 - 1.93T + 43T^{2} \) |
| 47 | \( 1 - 7.34T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 7.38T + 59T^{2} \) |
| 61 | \( 1 - 0.785T + 61T^{2} \) |
| 67 | \( 1 - 9.76T + 67T^{2} \) |
| 71 | \( 1 - 4.49T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 + 8.12T + 83T^{2} \) |
| 89 | \( 1 - 4.95T + 89T^{2} \) |
| 97 | \( 1 + 4.84T + 97T^{2} \) |
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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
−7.939232841256211093224997622224, −7.14001360130830865951227060752, −6.53839790195069832914069901804, −5.48236789813450688000266514365, −5.03594267123441789277846920680, −3.99370509137710146143833765794, −3.47810066055436030959778845766, −2.69630150123131372540183818876, −2.22663620354575294828191927765, −1.02178353308149769906646830319,
1.02178353308149769906646830319, 2.22663620354575294828191927765, 2.69630150123131372540183818876, 3.47810066055436030959778845766, 3.99370509137710146143833765794, 5.03594267123441789277846920680, 5.48236789813450688000266514365, 6.53839790195069832914069901804, 7.14001360130830865951227060752, 7.939232841256211093224997622224
