| L(s) = 1 | + 2.44·3-s − 2·7-s + 2.99·9-s − 4.44·11-s + 13-s − 6.89·17-s + 0.449·19-s − 4.89·21-s − 6.44·23-s + 4·29-s − 4.44·31-s − 10.8·33-s − 4.89·37-s + 2.44·39-s + 10.8·41-s − 11.3·43-s + 2·47-s − 3·49-s − 16.8·51-s − 1.10·53-s + 1.10·57-s + 9.34·59-s − 5.79·61-s − 5.99·63-s + 5.10·67-s − 15.7·69-s − 3.55·71-s + ⋯ |
| L(s) = 1 | + 1.41·3-s − 0.755·7-s + 0.999·9-s − 1.34·11-s + 0.277·13-s − 1.67·17-s + 0.103·19-s − 1.06·21-s − 1.34·23-s + 0.742·29-s − 0.799·31-s − 1.89·33-s − 0.805·37-s + 0.392·39-s + 1.70·41-s − 1.73·43-s + 0.291·47-s − 0.428·49-s − 2.36·51-s − 0.151·53-s + 0.145·57-s + 1.21·59-s − 0.742·61-s − 0.755·63-s + 0.623·67-s − 1.90·69-s − 0.421·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 4.44T + 11T^{2} \) |
| 17 | \( 1 + 6.89T + 17T^{2} \) |
| 19 | \( 1 - 0.449T + 19T^{2} \) |
| 23 | \( 1 + 6.44T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 4.44T + 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 1.10T + 53T^{2} \) |
| 59 | \( 1 - 9.34T + 59T^{2} \) |
| 61 | \( 1 + 5.79T + 61T^{2} \) |
| 67 | \( 1 - 5.10T + 67T^{2} \) |
| 71 | \( 1 + 3.55T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 - 4.89T + 79T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 7.79T + 97T^{2} \) |
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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
−8.390579709095279782253314366346, −8.016361451205750368657510633918, −7.08969235129423655510865365764, −6.34840803617623298887686004845, −5.32816859233799727954467697358, −4.26634528158627670201356391739, −3.47987180236557990998935249375, −2.62756458243053411118615905809, −2.00783520248972867229596459855, 0,
2.00783520248972867229596459855, 2.62756458243053411118615905809, 3.47987180236557990998935249375, 4.26634528158627670201356391739, 5.32816859233799727954467697358, 6.34840803617623298887686004845, 7.08969235129423655510865365764, 8.016361451205750368657510633918, 8.390579709095279782253314366346
