| L(s) = 1 | + 3·3-s − 8·4-s + 12·5-s + 9·9-s − 36·11-s − 24·12-s − 13·13-s + 36·15-s + 64·16-s + 78·17-s − 74·19-s − 96·20-s − 96·23-s + 19·25-s + 27·27-s + 18·29-s + 214·31-s − 108·33-s − 72·36-s − 286·37-s − 39·39-s + 384·41-s + 524·43-s + 288·44-s + 108·45-s − 300·47-s + 192·48-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1.07·5-s + 1/3·9-s − 0.986·11-s − 0.577·12-s − 0.277·13-s + 0.619·15-s + 16-s + 1.11·17-s − 0.893·19-s − 1.07·20-s − 0.870·23-s + 0.151·25-s + 0.192·27-s + 0.115·29-s + 1.23·31-s − 0.569·33-s − 1/3·36-s − 1.27·37-s − 0.160·39-s + 1.46·41-s + 1.85·43-s + 0.986·44-s + 0.357·45-s − 0.931·47-s + 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.335073301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.335073301\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 17 | \( 1 - 78 T + p^{3} T^{2} \) |
| 19 | \( 1 + 74 T + p^{3} T^{2} \) |
| 23 | \( 1 + 96 T + p^{3} T^{2} \) |
| 29 | \( 1 - 18 T + p^{3} T^{2} \) |
| 31 | \( 1 - 214 T + p^{3} T^{2} \) |
| 37 | \( 1 + 286 T + p^{3} T^{2} \) |
| 41 | \( 1 - 384 T + p^{3} T^{2} \) |
| 43 | \( 1 - 524 T + p^{3} T^{2} \) |
| 47 | \( 1 + 300 T + p^{3} T^{2} \) |
| 53 | \( 1 - 558 T + p^{3} T^{2} \) |
| 59 | \( 1 + 576 T + p^{3} T^{2} \) |
| 61 | \( 1 + 74 T + p^{3} T^{2} \) |
| 67 | \( 1 - 38 T + p^{3} T^{2} \) |
| 71 | \( 1 + 456 T + p^{3} T^{2} \) |
| 73 | \( 1 - 682 T + p^{3} T^{2} \) |
| 79 | \( 1 - 704 T + p^{3} T^{2} \) |
| 83 | \( 1 - 888 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1020 T + p^{3} T^{2} \) |
| 97 | \( 1 + 110 T + p^{3} T^{2} \) |
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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
−8.954388903432330627851021993782, −8.081163450822757355192204242845, −7.61508001583355464737563535836, −6.26673354113373740967222397635, −5.59931697557312560207961592632, −4.81289586777557833948334600696, −3.90915149751337652523023465009, −2.82806945656315480332186317919, −1.96262509147716991862856529259, −0.69192790621452876360891188318,
0.69192790621452876360891188318, 1.96262509147716991862856529259, 2.82806945656315480332186317919, 3.90915149751337652523023465009, 4.81289586777557833948334600696, 5.59931697557312560207961592632, 6.26673354113373740967222397635, 7.61508001583355464737563535836, 8.081163450822757355192204242845, 8.954388903432330627851021993782
