L(s) = 1 | + 4·3-s + 5·5-s + 4·7-s + 7·9-s + 20·15-s + 16·21-s − 44·23-s + 25·25-s − 8·27-s + 22·29-s + 20·35-s + 62·41-s − 76·43-s + 35·45-s + 4·47-s − 33·49-s + 58·61-s + 28·63-s + 116·67-s − 176·69-s + 100·75-s − 95·81-s − 76·83-s + 88·87-s − 142·89-s − 122·101-s − 44·103-s + ⋯ |
L(s) = 1 | + 4/3·3-s + 5-s + 4/7·7-s + 7/9·9-s + 4/3·15-s + 0.761·21-s − 1.91·23-s + 25-s − 0.296·27-s + 0.758·29-s + 4/7·35-s + 1.51·41-s − 1.76·43-s + 7/9·45-s + 4/47·47-s − 0.673·49-s + 0.950·61-s + 4/9·63-s + 1.73·67-s − 2.55·69-s + 4/3·75-s − 1.17·81-s − 0.915·83-s + 1.01·87-s − 1.59·89-s − 1.20·101-s − 0.427·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.015301371\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.015301371\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
good | 3 | \( 1 - 4 T + p^{2} T^{2} \) |
| 7 | \( 1 - 4 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 + 44 T + p^{2} T^{2} \) |
| 29 | \( 1 - 22 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( 1 - 62 T + p^{2} T^{2} \) |
| 43 | \( 1 + 76 T + p^{2} T^{2} \) |
| 47 | \( 1 - 4 T + p^{2} T^{2} \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 58 T + p^{2} T^{2} \) |
| 67 | \( 1 - 116 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( 1 + 76 T + p^{2} T^{2} \) |
| 89 | \( 1 + 142 T + p^{2} T^{2} \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−11.36557375328696613402873798062, −10.14242724505272250474643863845, −9.551201446942913015030291934024, −8.509081267817681463388950779282, −7.934971227553052378192820105367, −6.61069813333769198944009927879, −5.43604378190078653650128548413, −4.08047292942168149987346636493, −2.70961430055421932406734153548, −1.73738228925401918617574601968,
1.73738228925401918617574601968, 2.70961430055421932406734153548, 4.08047292942168149987346636493, 5.43604378190078653650128548413, 6.61069813333769198944009927879, 7.934971227553052378192820105367, 8.509081267817681463388950779282, 9.551201446942913015030291934024, 10.14242724505272250474643863845, 11.36557375328696613402873798062