| L(s) = 1 | + 1.87·2-s − 2.70·3-s + 1.52·4-s + 1.35·5-s − 5.08·6-s − 0.894·8-s + 4.33·9-s + 2.54·10-s − 3.15·11-s − 4.12·12-s + 6.06·13-s − 3.67·15-s − 4.72·16-s − 0.602·17-s + 8.13·18-s − 0.104·19-s + 2.06·20-s − 5.92·22-s − 0.943·23-s + 2.42·24-s − 3.16·25-s + 11.3·26-s − 3.60·27-s − 1.41·29-s − 6.89·30-s + 3.09·31-s − 7.08·32-s + ⋯ |
| L(s) = 1 | + 1.32·2-s − 1.56·3-s + 0.761·4-s + 0.606·5-s − 2.07·6-s − 0.316·8-s + 1.44·9-s + 0.804·10-s − 0.952·11-s − 1.19·12-s + 1.68·13-s − 0.947·15-s − 1.18·16-s − 0.146·17-s + 1.91·18-s − 0.0240·19-s + 0.461·20-s − 1.26·22-s − 0.196·23-s + 0.494·24-s − 0.632·25-s + 2.23·26-s − 0.694·27-s − 0.261·29-s − 1.25·30-s + 0.555·31-s − 1.25·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.170932856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.170932856\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 1.87T + 2T^{2} \) |
| 3 | \( 1 + 2.70T + 3T^{2} \) |
| 5 | \( 1 - 1.35T + 5T^{2} \) |
| 11 | \( 1 + 3.15T + 11T^{2} \) |
| 13 | \( 1 - 6.06T + 13T^{2} \) |
| 17 | \( 1 + 0.602T + 17T^{2} \) |
| 19 | \( 1 + 0.104T + 19T^{2} \) |
| 23 | \( 1 + 0.943T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 3.09T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 + 4.27T + 41T^{2} \) |
| 43 | \( 1 + 3.68T + 43T^{2} \) |
| 47 | \( 1 - 1.36T + 47T^{2} \) |
| 53 | \( 1 - 1.57T + 53T^{2} \) |
| 59 | \( 1 - 9.87T + 59T^{2} \) |
| 61 | \( 1 - 0.654T + 61T^{2} \) |
| 67 | \( 1 + 3.75T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 8.75T + 73T^{2} \) |
| 79 | \( 1 - 6.95T + 79T^{2} \) |
| 83 | \( 1 + 8.05T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 1.78T + 97T^{2} \) |
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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
−7.87137095824408489635218336944, −6.78728439273690630663121409616, −6.24521162337148701682447986827, −5.77881171616835052683550252559, −5.36964528515670271572246235276, −4.58097043774489021730652873366, −3.95232071045036348317111825909, −2.99085032422173507661751988522, −1.92107744556061185479820953118, −0.67578110410409153300447979060,
0.67578110410409153300447979060, 1.92107744556061185479820953118, 2.99085032422173507661751988522, 3.95232071045036348317111825909, 4.58097043774489021730652873366, 5.36964528515670271572246235276, 5.77881171616835052683550252559, 6.24521162337148701682447986827, 6.78728439273690630663121409616, 7.87137095824408489635218336944
